Statics bedford 5 chap 03

Problem 3.1 In Active Example 3.1, suppose that theangle between the ramp supporting the car is increasedfrom 20° to 30°. Draw the free-body diagram of the carshowing the new geometry. Suppose that the cable fromA to B must exert a 1900-lb horizontal force on the carto hold it in place. Determine the car’s weight in pounds.

A B

20�

Solution: The free-body diagram is shown to the right.Applying the equilibrium equations

∑Fx : T � N sin 30° D 0,

∑Fy : N cos 30° � mg D 0

Setting T D 1900 lb and solving yields

N D 3800 lb, mg D 3290 lb

Problem 3.2 The ring weighs 5 lb and is in equilib-rium. The force F1 D 4.5 lb. Determine the force F2 andthe angle ˛.

x

y

30�

F2

F1a

Solution: The free-body diagram is shown below the drawing. Theequilibrium equations are

∑Fx : F1 cos 30° � F2 cos ˛ D 0

∑Fy : F1 sin 30° C F2 sin ˛ � 5 lb D 0

We can write these equations as

F2 sin ˛ D 5 lb � F1 sin 30°

F2 cos ˛ D F1 cos 30°

Dividing these equations and using the known value for F1 we have.

tan ˛ D 5 lb � �4.5 lb� sin 30°

�4.5 lb� cos 30°D 0.706 ) ˛ D 35.2°

F2 D �4.5 lb� cos 30°

cos ˛D 4.77 lb

F2 D 4.77 lb, ˛ D 35.2°

84

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Problem 3.3 In Example 3.2, suppose that the attach-ment point C is moved to the right and cable AC isextended so that the angle between cable AC and theceiling decreases from 45° to 35°. The angle betweencable AB and the ceiling remains 60°. What are thetensions in cables AB and AC?

BC

A

60�45�

Solution: The free-body diagram is shown below the picture.The equilibrium equations are:

∑Fx : TAC cos 35° � TAB cos 60° D 0

∑Fy : TAC sin 35° C TAB sin 60° � 1962 N D 0

Solving we find

TAB D 1610 N, TAC D 985 N

Problem 3.4 The 200-kg engine block is suspendedby the cables AB and AC. The angle ˛ D 40°. The free-body diagram obtained by isolating the part of the systemwithin the dashed line is shown. Determine the forcesTAB and TAC.

BC

A Ax

y

a a

TAB TAC

(200 kg) (9.81 m/s2)

Solution:

˛ D 40°

∑Fx : TAC cos ˛ � TAB cos ˛ D 0

∑Fy : TAC sin ˛ C TAB sin ˛ � 1962 N D 0

Solving: TAB D TAC D 1.526 kN

TAB TAC

α α

1962 Ν


85

Problem 3.5 A heavy rope used as a mooring line fora cruise ship sags as shown. If the mass of the rope is90 kg, what are the tensions in the rope at A and B?

B 40�

55� A

Solution: The free-body diagram is shown.The equilibrium equations are

∑Fx : TB cos 40° � TA cos 55° D 0

∑Fy : TB sin 40° C TA sin 55° � 90�9.81� N D 0

Solving: TA D 679 N, TB D 508 N

Problem 3.6 A physiologist estimates that themasseter muscle of a predator, Martes, is capable ofexerting a force M as large as 900 N. Assume thatthe jaw is in equilibrium and determine the necessaryforce T that the temporalis muscle exerts and the forceP exerted on the object being bitten.

T

22�

P

M

36�

Solution: The equilibrium equations are

∑Fx : T cos 22° � M cos 36° D 0

∑Fy : T sin 22° C M sin 36° � P D 0

Setting M D 900 N, and solving, we find

T D 785 N, P D 823 N

86


Problem 3.7 The two springs are identical, with un-stretched lengths 250 mm and spring constants k D1200 N/m.

(a) Draw the free-body diagram of block A.(b) Draw the free-body diagram of block B.(c) What are the masses of the two blocks?

B

A

300 mm

280 mm

Solution: The tension in the upper spring acts on block A in thepositive Y direction, Solve the spring force-deflection equation forthe tension in the upper spring. Apply the equilibrium conditions toblock A. Repeat the steps for block B.

TUA D 0i C(

1200N

m

)�0.3 m � 0.25 m�j D 0i C 60j N

Similarly, the tension in the lower spring acts on block A in the nega-tive Y direction

TLA D 0i �(

1200N

m

)�0.28 m � 0.25 m�j D 0i � 36j N

The weight is WA D 0i � jWAjj

The equilibrium conditions are

∑F D

∑Fx C

∑Fy D 0,

∑F D WA C TUA C TLA D 0

Collect and combine like terms in i, j

∑Fy D ��jWAj C 60 � 36�j D 0

Solve jWAj D �60 � 36� D 24 N

The mass of A is

mA D jWL jjgj D 24 N

9.81 m/s2 D 2.45 kg

The free body diagram for block B is shown.

The tension in the lower spring TLB D 0i C 36j

The weight: WB D 0i � jWBjjApply the equilibrium conditions to block B.

∑F D WB C TLB D 0

Collect and combine like terms in i, j:

∑Fy D ��jWBj C 36�j D 0

Solve: jWBj D 36 N

The mass of B is given by mB D jWBjjgj D 36 N

9.81 m/s2 D 3.67 kg

B

A

300 mm

280 mm

A

Tension,upper spring

Tension,lowerspring

Weight,mass A

B

Tension,lower spring

Weight,mass B

x

y


87

Problem 3.8 The two springs in Problem 3.7 are iden-tical, with unstretched lengths of 250 mm. Suppose thattheir spring constant k is unknown and the sum of themasses of blocks A and B is 10 kg. Determine the valueof k and the masses of the two blocks.

Solution: All of the forces are in the vertical direction so we willuse scalar equations. First, consider the upper spring supporting bothmasses (10 kg total mass). The equation of equilibrium for block theentire assembly supported by the upper spring is A is TUA � �mA CmB�g D 0, where TUA D k��U � 0.25� N. The equation of equilibriumfor block B is TUB � mBg D 0, where TUB D k��L � 0.25� N. Theequation of equilibrium for block A alone is TUA C TLA � mAg D 0where TLA D �TUB. Using g D 9.81 m/s2, and solving simultane-

ously, we get k D 1962 N/m, mA D 4 kg, and mB D 6 kg .

Problem 3.9 The inclined surface is smooth (Remem-ber that “smooth” means that friction is negligble). Thetwo springs are identical, with unstretched lengths of250 mm and spring constants k D 1200 N/m. What arethe masses of blocks A and B?

B

A

300 mm

280 mm

30�

Solution:

F1 D �1200 N/m��0.3 � 0.25�m D 60 N

F2 D �1200 N/m��0.28 � 0.25�m D 36 N

∑FB &: �F2 C mBg sin 30° D 0

∑FA &: �F1 C F2 C mAg sin 30° D 0

Solving: mA D 4.89 kg, mB D 7.34 kg

mAg

mBg

F1

F2

F2

NA

NB

88


Problem 3.10 The mass of the crane is 20,000 kg. Thecrane’s cable is attached to a caisson whose mass is400 kg. The tension in the cable is 1 kN.

(a) Determine the magnitudes of the normal andfriction forces exerted on the crane by thelevel ground.

(b) Determine the magnitudes of the normal andfriction forces exerted on the caisson by thelevel ground.

Strategy: To do part (a), draw the free-body diagramof the crane and the part of its cable within thedashed line.

45°

Solution:

(a)∑

Fy : Ncrane � 196.2 kN � 1 kN sin 45° D 0

∑Fx : �Fcrane C 1 kN cos 45° D 0

Ncrane D 196.9 kN, Fcrane D 0.707 kN

(b)∑

Fy : Ncaisson � 3.924 kN C 1 kN sin 45° D 0

∑Fx : �1 kN cos 45° C Fcaisson D 0

Ncaisson D 3.22 kN, Fcaisson D 0.707 kN

Ncrane

Fcrane

196.2 kN1 kN

x

y

45°

45°

1 kN3.924 kN

Ncaisson

Fcaisson

Problem 3.11 The inclined surface is smooth. The100-kg crate is held stationary by a force T applied tothe cable.

(a) Draw the free-body diagram of the crate.(b) Determine the force T.

60�

T

Solution:

(a) The FBD

60° Ν

T

981 Ν

(b)∑

F -: T � 981 N sin 60° D 0

T D 850 N


89

Problem 3.12 The 1200-kg car is stationary on thesloping road.

(a) If ˛ D 20°, what are the magnitudes of the totalnormal and friction forces exerted on the car’s tiresby the road?

(b) The car can remain stationary only if the totalfriction force necessary for equilibrium is notgreater than 0.6 times the total normal force.What is the largest angle ˛ for which the car canremain stationary?

a

Solution:

(a) ˛ D 20°

∑F% : N � 11.772 kN cos ˛ D 0

∑F- : F � 11.772 kN sin ˛ D 0

N D 11.06 kN, F D 4.03 kN

(b) F D 0.6 N

∑F% : N � 11.772 kN cos ˛ D 0 ) ˛ D 31.0°

∑F- : F � 11.772 kN sin ˛ D 0

N

αF

11.772 kN

Problem 3.13 The 100-lb crate is in equilibrium on thesmooth surface. The spring constant is k D 400 lb/ft. LetS be the stretch of the spring. Obtain an equation for S(in feet) as a function of the angle ˛.

a

Solution: The free-body diagram is shown.The equilibrium equation in the direction parallel to the inclinedsurface is

kS � �100 lb� sin ˛ D 0

Solving for S and using the given value for k we find

S D �0.25 ft� sin ˛

90


Problem 3.14 A 600-lb box is held in place on thesmooth bed of the dump truck by the rope AB.

(a) If ˛ D 25°, what is the tension in the rope?(b) If the rope will safely support a tension of 400 lb,

what is the maximum allowable value of ˛?α

A

B

Solution: Isolate the box. Resolve the forces into scalar compo-nents, and solve the equilibrium equations.

The external forces are the weight, the tension in the rope, and thenormal force exerted by the surface. The angle between the x axis andthe weight vector is ��90 � ˛� (or 270 C ˛). The weight vector is

W D jWj�i sin ˛ � j cos ˛� D �600��i sin ˛ � j cos ˛�

The projections of the rope tension and the normal force are

T D �jTxji C 0j N D 0i C jNy jj


∑F D W C N C T D 0

Substitute, and collect like terms

∑Fx D �600 sin ˛ � jTx j�i D 0

∑Fy D ��600 cos ˛ C jNy j�j D 0

Solve for the unknown tension when

For ˛ D 25°

jTxj D 600 sin ˛ D 253.6 lb.

For a tension of 400 lb, (600 sin ˛ � 400� D 0. Solve for the unknownangle

sin ˛ D 400

600D 0.667 or ˛ D 41.84°

α

AB

T

N

W

y

x

α


91

Problem 3.15 The 80-lb box is held in place onthe smooth inclined surface by the rope AB. Determinethe tension in the rope and the normal force exertedon the box by the inclined surface.

A

B30�

50�

Solution: The equilibrium equations (in terms of a coordinatesystem with the x axis parallel to the inclined surface) are

∑Fx : �80 lb� sin 50° � T cos 50 D 0

∑Fx : N � �80 lb� cos 50° � T sin 50 D 0

Solving: T D 95.34 lb, N D 124 lb

Problem 3.16 The 1360-kg car and the 2100-kg towtruck are stationary. The muddy surface on which thecar’s tires rest exerts negligible friction forces on them.What is the tension in the tow cable?

18�10�

26�

Solution: FBD of the car being towed

∑F- : T cos 8° � 13.34 kN sin 26° D 0

T D 5.91 kN

13.34 kN

T

N

18°

26°

92


Problem 3.17 Each box weighs 40 lb. The angles aremeasured relative to the horizontal. The surfaces aresmooth. Determine the tension in the rope A and thenormal force exerted on box B by the inclined surface.

D

C

B

A

70�

20�

45�

Solution: The free-body diagrams are shown.The equilibrium equations for box D are

∑Fx : �40 lb� sin 20° � TC cos 25° D 0

∑Fy : ND � �40 lb� cos 20° C TC sin 25° D 0

The equilibrium equations for box B are

∑Fx : �40 lb� sin 70° C TC cos 25° � TA D 0

∑Fy : NB � �40 lb� cos 70° C TC sin 25° D 0

Solving these four equations yields:

TA D 51.2 lb, TC D 15.1 lb, NB D 7.30 lb, ND D 31.2 lb

Thus TA D 51.2 lb, NB D 7.30 lb

Problem 3.18 A 10-kg painting is hung with a wiresupported by a nail. The length of the wire is 1.3 m.

(a) What is the tension in the wire?(b) What is the magnitude of the force exerted on the

nail by the wire? 1.2 m

Solution:

(a)∑

Fy : 98.1 N � 25

13T D 0

T D 128 N

(b) Force D 98.1 N

98.1 N

T T5

12 12


93

Problem 3.19 A 10-kg painting is hung with a wiresupported by two nails. The length of the wire is 1.3 m.

(a) What is the tension in the wire?(b) What is the magnitude of the force exerted on each

nail by the wire? (Assume that the tension is thesame in each part of the wire.)

Compare your answers to the answers to Problem 3.18.

0.4 m0.4 m 0.4 m

Solution:

(a) Examine the point on the left where the wire is attached to thepicture. This point supports half of the weight

∑Fy : T sin 27.3° � 49.05 N D 0

T D 107 N

(b) Examine one of the nails

∑Fx : �Rx � T cos 27.3° C T D 0

∑Fy : Ry � T sin 27.3° D 0

R D√

Rx2 C Ry

2

R D 50.5 N

27.3°R

T

49.05 N

Ry

Rx

T

T

27.3°

Problem 3.20 Assume that the 150-lb climber is inequilibrium. What are the tensions in the rope on theleft and right sides?

14� 15�

Solution:

∑Fx D TR cos�15°� � TL cos�14°� D 0

∑Fy D TR sin�15°� C TL sin�14°� � 150 D 0

Solving, we get TL D 299 lb, TR D 300 lb

y

x

TRTL

14° 15°

150 lb

94


Problem 3.21 If the mass of the climber shown inProblem 3.20 is 80 kg, what are the tensions in the ropeon the left and right sides?

Solution:

∑Fx D TR cos�15°� � TL cos�14°� D 0

∑Fy D TR sin�15°� C TL sin�14°� � mg D 0

Solving, we get

TL D 1.56 kN, TR D 1.57 kN

14° 15°TL

TR

y

x

mg = (80) (9.81) N

Problem 3.22 The construction worker exerts a 20-lbforce on the rope to hold the crate in equilibrium in theposition shown. What is the weight of the crate?

5�

30�

Solution: The free-body diagram is shown.The equilibrium equations for the part of the rope system where thethree ropes are joined are

∑Fx : �20 lb� cos 30° � T sin 5° D 0

∑Fy : ��20 lb� sin 30° C T cos 5° � W D 0

Solving yields W D 188 lb


95

Problem 3.23 A construction worker on the moon,where the acceleration due to gravity is 1.62 m/s2, holdsthe same crate described in Problem 3.22 in the positionshown. What force must she exert on the cable to holdthe crate ub equilibrium (a) in newtons; (b) in pounds?

5�

30�

Solution: The free-body diagram is shown.From Problem 3.22 we know that the weight is W D 188 lb. Thereforeits mass is

m D 188 lb

32.2 ft/s2 D 5.84 slug

m D 5.84 slug

(14.59 kg

slug

)D 85.2 kg

The equilibrium equations for the part of the rope system where thethree ropes are joined are

∑Fx : F cos 30° � T sin 5° D 0

∑Fy : �F sin 30° C T cos 5° � mgm D 0

where gm D 1.62 m/s2.Solving yields F D 3.30 lb D 14.7 N

�a� F D 14.7 N, �b� F D 3.30 lb

96


Problem 3.24 The person wants to cause the 200-lbcrate to start sliding toward the right. To achieve this,the horizontal component of the force exerted on thecrate by the rope must equal 0.35 times the normal forceexerted on the crate by the floor. In Fig.a, the personpulls on the rope in the direction shown. In Fig.b, theperson attaches the rope to a support as shown and pullsupward on the rope. What is the magnitude of the forcehe must exert on the rope in each case?

10�

20�

(a)

(b)

Solution: The friction force Ffr is given by

Ffr D 0.35N

(a) For equilibrium we have

∑Fx : T cos 20° � 0.35N D 0

∑Fy : T sin 20° � 200 lb C N D 0

Solving: T D 66.1 lb

(b) The person exerts the force F. Using the free-body diagram ofthe crate and of the point on the rope where the person grabs therope, we find

∑Fx : TL � 0.35N D 0

∑Fy : N � 200 lb D 0

∑Fx : �TL C TR cos 10° D 0

∑Fy : F � TR sin 10° D 0

Solving we find F D 12.3 lb


97

Problem 3.25 A traffic engineer wants to suspend a200-lb traffic light above the center of the two rightlanes of a four-lane thoroughfare as shown. Determinethe tensions in the cables AB and BC.

30 ft

CA

B10 ft

20 ft80 ft

Solution:∑

Fx : � 6p37

TAB C 2p5

TBC D 0

∑Fy :

1p37

TAB C 1p5

TBC � 200 lb D 0

Solving: TAB D 304 lb, TBC D 335 lb

61

2

1

TBC

TAB

200 lb

Problem 3.26 Cable AB is 3 m long and cable BC is4 m long. The mass of the suspended object is 350 kg.Determine the tensions in cables AB and BC.

C

B

A

5m

Solution:∑

Fx : � 3

5TAB C 4

5TBC D 0

∑Fy :

4

5TAB C 3

5TBC � 3.43 kN D 0

TAB D 2.75 kN, TBC D 2.06 kN

4

4

33

TAB

TAC

3.43 kN

98


Problem 3.27 In Problem 3.26, the length of cable ABis adjustable. If you don’t want the tension in either cableAB or cable BC to exceed 3 kN, what is the minimumacceptable length of cable AB?

Solution: Consider the geometry:

We have the constraints

LAB2 D x2 C y2, �4 m�2 D �5 m � x�2 C y2

These constraint imply

y D√

�10 m�x � x2 � 9 m2

L D√

�10 m�x � 9 m2

Now draw the FBD and write the equations in terms of x

∑Fx : � xp

10x � 9TAB C 5 � x

4TBC D 0

∑Fy :

p10x � x2 � 9p

10x � 9TAB C

p10x � x2 � 9

4TBC � 3.43 kN D 0

If we set TAB D 3 kN and solve for x we find x D 1.535, TBC D2.11 kN < 3 kN

Using this value for x we find that LAB D 2.52 m

x

y

5−x

4 mLAB

TAB

TBC

x5−x

yy

4

3.43 kN


99

Problem 3.28 What are the tensions in the upper andlower cables? (Your answers will be in terms of W.Neglect the weight of the pulley.)

45° 30°

W

Solution: Isolate the weight. The frictionless pulley changes thedirection but not the magnitude of the tension. The angle between theright hand upper cable and the x axis is ˛, hence

TUR D jTUj�i cos ˛ C j sin ˛�.

The angle between the positive x and the left hand upper pulley is�180° � ˇ�, hence

TUL D jTUj�i cos�180 � ˇ� C j sin�180 � ˇ��

D jTUj��i cos ˇ C j sin ˇ�.

The lower cable exerts a force: TL D �jTLji C 0j

The weight: W D 0i � jWjj


∑F D W C TUL C TUR C TL D 0

Substitute and collect like terms,

∑Fx D ��jTUj cos ˇ C jTUj cos ˛ � jTL j�i D 0

∑Fy D �jTUj sin ˛ C jTUj sin ˇ � jWj�j D 0.

Solve: jTUj D( jWj

�sin ˛ C sin ˇ�

),

jTL j D jTUj�cos ˛ � cos ˇ�.

From which jTL j D jWj(

cos ˛ � cos ˇ

sin ˛ C sin ˇ

).

For ˛ D 30° and ˇ D 45°

jTUj D 0.828jWj,

jTL j D 0.132jWj

TU

TL

TU

y

x

W

β α

100


Problem 3.29 Two tow trucks lift a 660-lb motorcycleout of a ravine following an accident. If the motorcycleis in equilibrium in the position shown, what are thetensions in cables AB and AC? B

A

C(12, 32) ft

(36, 36) ft

(26, 16) ft

x

y

Solution: The angles are

˛ D tan�1(

32 � 16

26 � 12

)D 48.8°

ˇ D tan�1(

36 � 16

36 � 26

)D 63.4°

Now from equilibrium we have

∑Fx : �TAB cos ˛ C TAC cos ˇ D 0

∑Fy : TAB sin ˛ C TAC sin ˇ � 660 lb D 0

Solving yields TAB D 319 lb, TAC D 470 lb


101

Problem 3.30 An astronaut candidate conducts exper-iments on an airbearing platform. While she carries outcalibrations, the platform is held in place by the hori-zontal tethers AB, AC, and AD. The forces exerted bythe tethers are the only horizontal forces acting on theplatform. If the tension in tether AC is 2 N, what are thetensions in the other two tethers?

3.0 m 1.5 m

B C

TOP VIEW

D

4.0 m

3.5 m

A

Solution: Isolate the platform. The angles ˛ and ˇ are

tan ˛ D(

1.5

3.5

)D 0.429, ˛ D 23.2°.

Also, tan ˇ D(

3.0

3.5

)D 0.857, ˇ D 40.6°.

The angle between the tether AB and the positive x axis is �180° � ˇ�,hence

TAB D jTABj�i cos�180° � ˇ� C j sin�180° � ˇ��

TAB D jTABj��i cos ˇ C j sin ˇ�.

The angle between the tether AC and the positive x axis is �180° C ˛�.The tension is

TAC D jTACj�i cos�180° C ˛� C j sin�180° C ˛��

D jTACj��i cos ˛ � j sin ˛�.

The tether AD is aligned with the positive x axis, TAD D jTADji C 0j.

The equilibrium condition:

∑F D TAD C TAB C TAC D 0.


∑Fx D ��jTABj cos ˇ � jTACj cos ˛ C jTADj�i D 0,

∑Fy D �jTABj sin ˇ � jTACj sin ˛�j D 0.

3.5m

1.5 m

3.0 m

4.0m

A

B

C

D

B

β

αA

C

D

x

y

Solve: jTABj D(

sin ˛

sin ˇ

)jTACj,

jTADj D( jTACj sin�˛ C ˇ�

sin ˇ

).

For jTACj D 2 N, ˛ D 23.2° and ˇ D 40.6°,

jTABj D 1.21 N, jTADj D 2.76 N

102


Problem 3.31 The bucket contains concrete andweighs 5800 lb. What are the tensions in the cables ABand AC?

B C

A

(20, 34) ft(5, 34) ft

(12, 16) ft

y

xSolution: The angles are

˛ D tan�1(

34 � 16

12 � 5

)D 68.7°

ˇ D tan�1(

34 � 16

20 � 12

)D 66.0°

Now from equilibrium we have

∑Fx : �TAB cos ˛ C TAC cos ˇ D 0

∑Fy : TAB sin ˛ C TAC sin ˇ � 660 lb D 0

Solving yields TAB D 319 lb, TAC D 470 lb

Problem 3.32 The slider A is in equilibrium and thebar is smooth. What is the mass of the slider? 20�

45�

200 NA

Solution: The pulley does not change the tension in the rope thatpasses over it. There is no friction between the slider and the bar.

Eqns. of Equilibrium:

∑Fx D T sin 20° C N cos 45° D 0 �T D 200 N�

∑Fy D N sin 45° C T cos 20° � mg D 0 g D 9.81 m/s2

Substituting for T and g, we have two eqns in two unknowns(N and m).

Solving, we get N D �96.7 N, m D 12.2 kg.

y

45°

20°

N

x

T = 200 N

mg = (9.81) g


103

Problem 3.33 The 20-kg mass is suspended from threecables. Cable AC is equipped with a turnbuckle so thatits tension can be adjusted and a strain gauge that allowsits tension to be measured. If the tension in cable AC is40 N, what are the tensions in cables AB and AD?

B C

A

D

0.4 m0.4 m 0.48 m

0.64 m

Solution:

TAC D 40 N

∑Fx : � 5p

89TAB C 5p

89TAC C 11p

185TAD D 0

∑Fy :

8p89

TAB C 8p89

TAC C 8p185

TAD � 196.2 N D 0

Solving: TAB D 144.1 N, TAD D 68.2 N

TAB

TAC

TAD

11

88 8

5

5

196.2 N

Problem 3.34 The structural joint is in equilibrium. IfFA D 1000 lb and FD D 5000 lb, what are FB and FC?

FB

FC

FA FD

80�

65�

35�


∑Fx : FD � FC cos 65° � FB cos 35° � FA D 0

∑Fy : �FC sin 65° C FB sin 35° D 0

Solving yields FB D 3680 lb, FC D 2330 lb

104


Problem 3.35 The collar A slides on the smoothvertical bar. The masses mA D 20 kg and mB D 10 kg.When h D 0.1 m, the spring is unstretched. When thesystem is in equilibrium, h D 0.3 m. Determine thespring constant k.

B

A

h

k

0.25 m

Solution: The triangles formed by the rope segments and the hori-zontal line level with A can be used to determine the lengths Lu andLs. The equations are

Lu D√

�0.25�2 C �0.1�2 and Ls D√

�0.25�2 C �0.3�2.

The stretch in the spring when in equilibrium is given by υ D Ls � Lu.Carrying out the calculations, we get Lu D 0.269 m, Ls D 0.391 m,and υ D 0.121 m. The angle, �, between the rope at A and the hori-zontal when the system is in equilibrium is given by tan � D 0.3/0.25,or � D 50.2°. From the free body diagram for mass A, we get twoequilibrium equations. They are

∑Fx D �NA C T cos � D 0

and∑

Fy D T sin � � mAg D 0.

We have two equations in two unknowns and can solve. We get NA D163.5 N and T D 255.4 N. Now we go to the free body diagram for B,where the equation of equilibrium is T � mBg � kυ D 0. This equation

has only one unknown. Solving, we get k D 1297 N/m

Lu

LuLs

0.1 m

0.3 m

0.25 m

0.25 m

NA A

T

mAg

mBg

T

B

Kδ


105

Problem 3.36* Suppose that you want to design acable system to suspend an object of weight W fromthe ceiling. The two wires must be identical, and thedimension b is fixed. The ratio of the tension T in eachwire to its cross-sectional area A must equal a specifiedvalue T/A D �. The “cost” of your design is the totalvolume of material in the two wires, V D 2A

pb2 C h2.

Determine the value of h that minimizes the cost.

W

b

h

b

Solution: From the equation

∑Fy D 2T sin � � W D 0,

we obtain T D W

2 sin �D W

pb2 C h2

2h.

Since T/A D �, A D T

�D W

pb2 C h2

2�h

and the “cost” is V D 2Ap

b2 C h2 D W�b2 C h2�

�h.

To determine the value of h that minimizes V, we set

dV

dhD W

�

[� �b2 C h2�

h2C 2

]D 0

and solve for h, obtaining h D b.

T T

W

θ θ

106


Problem 3.37 The system of cables suspends a1000-lb bank of lights above a movie set. Determinethe tensions in cables AB, CD, and CE.

D

C

A

E

45° 30°

20 ft

B

18 ft

Solution: Isolate juncture A, and solve the equilibrium equations.Repeat for the cable juncture C.

The angle between the cable AC and the positive x axis is ˛. Thetension in AC is TAC D jTACj�i cos ˛ C j sin ˛�

The angle between the x axis and AB is �180° � ˇ�. The tension is

TAB D jTABj�i cos�180 � ˇ� C j sin�180 � ˇ��

TAB D ��i cos ˇ C j sin ˇ�.

The weight is W D 0i � jWjj.


∑F D 0 D W C TAB C TAC D 0.


∑Fx D �jTACj cos ˛ � jTABj cos ˇ�i D 0

∑Fy D �jTABj sin ˇ C jTACj sin ˛ � jWj�j D 0.

Solving, we get

jTABj D(

cos ˛

cos ˇ

)jTACj and jTACj D

( jWj cos ˇ

sin�˛ C ˇ�

),

jWj D 1000 lb, and ˛ D 30°, ˇ D 45°

jTACj D �1000�

(0.7071

0.9659

)D 732.05 lb

jTABj D �732�

(0.866

0.7071

)D 896.5 lb

Isolate juncture C. The angle between the positive x axis and the cableCA is �180° � ˛�. The tension is

TCA D jTCAj�i cos�180° C ˛� C j sin�180° C ˛��,

or TCA D jTCAj��i cos ˛ � j sin ˛�.

The tension in the cable CE is

TCE D ijTCEj C 0j.

The tension in the cable CD is TCD D 0i C jjTCDj.


∑F D 0 D TCA C TCE C TCD D 0

Substitute t and collect like terms,

∑Fx D �jTCEj � jTCAj cos ˛�i D 0,

∑Fy D �jTCDj � jTCAj sin ˛�j D 0.

Solve: jTCEj D jTCAj cos ˛,

jTCDj D jTCAj sin ˛;

for jTCAj D 732 lb and ˛ D 30°,

jTABj D 896.6 lb,

jTCEj D 634 lb,

jTCDj D 366 lb

BC

A

W

y

x

β α

D

C

Aα

E

y

x

90°


107

Problem 3.38 Consider the 1000-lb bank of lights inProblem 3.37. A technician changes the position of thelights by removing the cable CE. What is the tension incable AB after the change?

Solution: The original configuration in Problem 3.35 is used tosolve for the dimensions and the angles. Isolate the juncture A, andsolve the equilibrium conditions.

The lengths are calculated as follows: The vertical interior distancein the triangle is 20 ft, since the angle is 45 deg. and the base andaltitude of a 45 deg triangle are equal. The length AB is given by

AB D 20 ft

cos 45°D 28.284 ft.

The length AC is given by

AC D 18 ft

cos 30°D 20.785 ft.

The altitude of the triangle for which AC is the hypotenuse is18 tan 30° D 10.392 ft. The distance CD is given by 20 � 10.392 D9.608 ft.

The distance AD is given by

AD D AC C CD D 20.784 C 9.608 D 30.392

The new angles are given by the cosine law

AB2 D 382 C AD2 � 2�38��AD� cos ˛.

Reduce and solve:

cos ˛ D(

382 C �30.392�2 � �28.284�2

2�38��30.392�

)D 0.6787, ˛ D 47.23°.

cos ˇ D(

�28.284�2 C �38�2 � �30.392�2

2�28.284��38�

)D 0.6142, ˇ D 52.1°.

Isolate the juncture A. The angle between the cable AD and the positivex axis is ˛. The tension is:

TAD D jTADj�i cos ˛ C j sin ˛�.

The angle between x and the cable AB is �180° � ˇ�. The tension is

TAB D jTABj��i cos ˇ C j sin ˇ�.

The weight is W D 0i � jWjj


∑F D 0 D W C TAB C TAD D 0.


∑Fx D �jTADj cos ˛ � jTABj cos ˇ�i D 0,

∑Fy D �jTABj sin ˇ C jTADj sin ˛ � jWj�j D 0.

20 ft 18 ft

BC

A

D

αβ

B

B

W

y

x

A

A

D

D

38

β α

α

β

β

α

28.284 20.784+9.608= 30.392

Solve: jTABj D(

cos ˛

cos ˇ

)jTADj,

and jTADj D( jWj cos ˇ

sin�˛ C ˇ�

).

For jWj D 1000 lb, and ˛ D 51.2°, ˇ D 47.2°

jTADj D �1000�

(0.6142

0.989

)D 621.03 lb,

jTABj D �622.3�

(0.6787

0.6142

)D 687.9 lb

108


Problem 3.39 While working on another exhibit, acurator at the Smithsonian Institution pulls the suspendedVoyager aircraft to one side by attaching three horizontalcables as shown. The mass of the aircraft is 1250 kg.Determine the tensions in the cable segments AB, BC,and CD.

A

B

C

D

70°

50°

30°

Solution: Isolate each cable juncture, beginning with A and solvethe equilibrium equations at each juncture. The angle between thecable AB and the positive x axis is ˛ D 70°; the tension in cable ABis TAB D jTABj�i cos ˛ C j sin ˛�. The weight is W D 0i � jWjj. Thetension in cable AT is T D �jTji C 0j. The equilibrium conditions are

∑F D W C T C TAB D 0.

Substitute and collect like terms

∑Fx�jTABj cos ˛ � jTj�i D 0,

∑Fy D �jTABj sin ˛ � jWj�j D 0.

Solve: the tension in cable AB is jTABj D( jWj

sin ˛

).

For jWj D �1250 kg�(

9.81m

s2

)D 12262.5 N and ˛ D 70°

jTABj D(

12262.5

0.94

)D 13049.5 N

Isolate juncture B. The angles are ˛ D 50°, ˇ D 70°, and the tensioncable BC is TBC D jTBCj�i cos ˛ C j sin ˛�. The angle between thecable BA and the positive x axis is �180 C ˇ�; the tension is

TBA D jTBAj�i cos�180 C ˇ� C j sin�180 C ˇ��

D jTBAj��i cos ˇ � j sin ˇ�

The tension in the left horizontal cable is T D �jTji C 0j. The equi-librium conditions are

∑F D TBA C TBC C T D 0.


∑Fx D �jTBCj cos ˛ � jTBAj cos ˇ � jTj�i D 0

∑Fy D �jTBCj sin ˛ � jTBAj sin ˇ�j D 0.

Solve: jTBCj D(

sin ˇ

sin ˛

)jTBAj.

For jTBAj D 13049.5 N, and ˛ D 50°, ˇ D 70°,

jTBCj D �13049.5�

(0.9397

0.7660

)D 16007.6 N

Isolate the cable juncture C. The angles are ˛ D 30°, ˇ D 50°. Bysymmetry with the cable juncture B above, the tension in cable CD is

jTCDj D(

sin ˇ

sin ˛

)jTCBj.

Substitute: jTCDj D �16007.6�

(0.7660

0.5

)D 24525.0 N.

This completes the problem solution.

y

x

TB

A

C

α

β

y

x

T

W

B

A α

y

x

C

B

T

D

α

β


109

Problem 3.40 A truck dealer wants to suspend a 4000-kg truck as shown for advertising. The distance b D15 m, and the sum of the lengths of the cables AB andBC is 42 m. Points A and C are at the same height. Whatare the tensions in the cables?

CAb

40 m

B

Solution: Determine the dimensions and angles of the cables. Iso-late the cable juncture B, and solve the equilibrium conditions. Thedimensions of the triangles formed by the cables:

b D 15 m, L D 25 m, AB C BC D S D 42 m.

Subdivide into two right triangles with a common side of unknownlength. Let the unknown length of this common side be d, then by thePythagorean Theorem b2 C d2 D AB2, L2 C d2 D BC2.

Subtract the first equation from the second to eliminate the unknown d,L2 � b2 D BC2 � AB2.

Note that BC2 � AB2 D �BC � AB��BC C AB�.

Substitute and reduce to the pair of simultaneous equations in theunknowns

BC � AB D(

L2 � b2

S

), BC C AB D S

Solve: BC D(

1

2

) (L2 � b2

SC S

)

D(

1

2

) (252 � 152

42C 42

)D 25.762 m

and AB D S � BC D 42 � 25.762 D 16.238 m.

The interior angles are found from the cosine law:

cos ˛ D(

�L C b�2 C BC2 � AB2

2�L C b��BC�

)D 0.9704 ˛ D 13.97°

cos ˇ D(

�L C b�2 C AB2 � BC2

2�L C b��AB�

)D 0.9238 ˇ D 22.52°

Isolate cable juncture B. The angle between BC and the positive x axisis ˛; the tension is

TBC D jTBCj�i cos ˛ C j sin ˛�

The angle between BA and the positive x axis is �180° � ˇ�; thetension is

TBA D jTBAj�i cos�180 � ˇ� C j sin�180 � ˇ��

D jTBAj��i cos ˇ C j sin ˇ�.

The weight is W D 0i � jWjj.


∑F D W C TBA C TBC D 0.

15 m 25 m

bA

A

C

B

B

W

C

y

x

L

α

α

β

β


∑Fx D �jTBCj cos ˛ � jTBAj cos ˇ�i D 0,

∑Fy D �jTBCj sin ˛ C jTBAj sin ˇ � jWj�j D 0

Solve: jTBCj D(

cos ˇ

cos ˛

)jTBAj,

and jTBAj D( jWj cos ˛

sin�˛ C ˇ�

).

For jWj D �4000��9.81� D 39240 N,

and ˛ D 13.97°, ˇ D 22.52°,

jTBAj D 64033 D 64 kN,

jTBCj D 60953 D 61 kN

110


Problem 3.41 The distance h D 12 in, and the tensionin cable AD is 200 lb. What are the tensions in cablesAB and AC?

12 in.

12 in.

12 in.

8 in.

8 in.h

D

B

A

C

Solution: Isolated the cable juncture. From the sketch, the anglesare found from

tan ˛ D(

8

12

)D 0.667 ˛ D 33.7°

tan ˇ D(

4

12

)D 0.333 ˇ D 18.4°

The angle between the cable AB and the positive x axis is �180° � ˛�,the tension in AB is:

TAB D jTABj�i cos�180 � ˛� C j sin�180 � ˛��

TAB D jTABj��i cos ˛ C j sin ˛�.

The angle between AC and the positive x axis is �180 C ˇ�. Thetension is

TAC D jTACj�i cos�180 C ˇ� C j sin�180 C ˇ��

TAC D jTACj��i cos ˇ � j sin ˇ�.

The tension in the cable AD is

TAD D jTADji C 0j.


∑F D TAC C TAB C TAD D 0.


∑Fx D ��jTABj cos ˛ � jTACj cos ˇ C jTADj�i D 0

∑Fy D �jTABj sin ˛ � jTACj sin ˇ�j D 0.

Solve: jTABj D(

sin ˇ

sin ˛

)jTACj,

and jTACj D(

sin ˛

sin�˛ C ˇ�

)jTADj.

For jTADj D 200 lb, ˛ D 33.7°, ˇ D 18.4°

jTACj D 140.6 lb, jTABj D 80.1 lb

y

x

C

D

B

A

α

β

12 in

8 in

4 in


111

Problem 3.42 You are designing a cable system tosupport a suspended object of weight W. Because yourdesign requires points A and B to be placed as shown,you have no control over the angle ˛, but you can choosethe angle ˇ by placing point C wherever you wish. Showthat to minimize the tensions in cables AB and BC, youmust choose ˇ D ˛ if the angle ˛ ½ 45°.

Strategy: Draw a diagram of the sum of the forcesexerted by the three cables at A.

W

CB

A

βα

Solution: Draw the free body diagram of the knot at point A. Thendraw the force triangle involving the three forces. Remember that ˛ isfixed and the force W has both fixed magnitude and direction. Fromthe force triangle, we see that the force TAC can be smaller thanTAB for a large range of values for ˇ. By inspection, we see that theminimum simultaneous values for TAC and TAB occur when the twoforces are equal. This occurs when ˛ D ˇ. Note: this does not happenwhen ˛ < 45°.

In this case, we solved the problem without writing the equations ofequilibrium. For reference, these equations are:

∑Fx D �TAB cos ˛ C TAC cos ˇ D 0

and∑

Fy D TAB sin ˛ C TAC sin ˇ � W D 0.

W

A

B

x

y

α

TAB

TAC

Possible locationsfor C lie on line

B

W

α

Candidate β

C? C?

TAB

Candidate valuesfor TAC

Fixed direction forline AB

Problem 3.43* The length of the cable ABC is 1.4 m.The 2-kN force is applied to a small pulley. The systemis stationary. What is the tension in the cable?

C

B

A1 m

0.75 m

15�2 kN

Solution: Examine the geometry

√h2 C �0.75 m�2 C

√h2 C �0.25 m�2 D 1.4 m

tan ˛ D h

0.75 m, tan ˇ D h

0.25 m

) h D 0.458 m, ˛ D 31.39°, ˇ D 61.35°

Now draw a FBD and solve for the tension. We can use either of theequilibrium equations

∑Fx : �T cos ˛ C T cos ˇ C �2 kN� sin 15° D 0

T D 1.38 kN

βα0.75 m 0.25 m

h

βα

2 kN

15°

T

T

112


Problem 3.44 The masses m1 D 12 kg and m2 D 6 kgare suspended by the cable system shown. The cable BCis horizontal. Determine the angle ˛ and the tensions inthe cables AB, BC, and CD.

C

A

αB

D

70�

m2

m1

117.7 N

TAB

TBCα

B

Solution: We have 4 unknowns and 4 equations

∑FBx : �TAB cos ˛ C TBC D 0

∑FBy : TAB sin ˛ � 117.7 N D 0

∑FCx : �TBC C TCD cos 70° D 0

∑FCy : TCD sin 70° � 58.86 N D 0

Solving we find

˛ D 79.7°, TAB D 119.7 N, TBC D 21.4 N, TCD D 62.6 N

70°

TCD

TBC

58.86 N

C

Problem 3.45 The weights W1 D 50 lb and W2 aresuspended by the cable system shown. Determine theweight W2 and the tensions in the cables AB, BC,and CD.

A D

BC

W1

W2

30 in 30 in 30 in

20 in 16 in

Solution: We have 4 unknowns and 4 equilibrium equations to use

∑FBx : � 3p

13TAB C 15p

229TBC D 0

∑FBy :

2p13

TAB C 2p229

TBC � 50 lb D 0

∑FCx : � 15p

229TBC C 15

17TCD D 0

∑FCy : � 2p

229TBC C 8

17TCD � W2 D 0

)W2 D 25 lb, TAB D 75.1 lb

TBC D 63.1 lb, TCD D 70.8 lb

TAB

TBC

50 lb

B

2

32

15

TCD

TBC

W2

2

15

8

15

C


113

Problem 3.46 In the system shown in Problem 3.45,assume that W2 D W1/2. If you don’t want the tensionanywhere in the supporting cable to exceed 200 lb, whatis the largest acceptable value of W1?

Solution:∑

FBx : � 3p13

TAB C 15p229

TBC D 0

∑FBy :

2p13

TAB C 2p229

TBC � W1 D 0

∑FCx : � 15p

229TBC C 15

17TCD D 0

∑FCy : � 2p

229TBC C 8

17TCD � W1

2D 0

TAB D 1.502W1, TBC D 1.262W1, TCD D 1.417W1

AB is the critical cable

200 lb D 1.502W1 ) W1 D 133.2 lb

W1

TAB

TBC

B

2

215

3

C

TCD

W2 = W1/2

15

2 15

8

TBC

114


Problem 3.47 The hydraulic cylinder is subjected tothree forces. An 8-kN force is exerted on the cylinderat B that is parallel to the cylinder and points from Btoward C. The link AC exerts a force at C that is parallelto the line from A to C. The link CD exerts a force atC that is parallel to the line from C to D.

(a) Draw the free-body diagram of the cylinder. (Thecylinder’s weight is negligible).

(b) Determine the magnitudes of the forces exerted bythe links AC and CD.

1 m

0.6 m Scoop

A B

D

C

0.15 m

0.6 m

1 m

Hydraulic�cylinder

Solution: From the figure, if C is at the origin, then points A, B,and D are located at

A�0.15, �0.6�

B�0.75, �0.6�

D�1.00, 0.4�

and forces FCA, FBC, and FCD are parallel to CA, BC, and CD, respec-tively.

We need to write unit vectors in the three force directions and expressthe forces in terms of magnitudes and unit vectors. The unit vectorsare given by

eCA D rCA

jrCAj D 0.243i � 0.970j

eCB D rCB

jrCBj D 0.781i � 0.625j

eCD D rCD

jrCDj D 0.928i C 0.371j

Now we write the forces in terms of magnitudes and unit vectors. Wecan write FBC as FCB D �8eCB kN or as FCB D 8��eCB� kN (becausewe were told it was directed from B toward C and had a magnitudeof 8 kN. Either way, we must end up with

FCB D �6.25i C 5.00j kN

Similarly,

FCA D 0.243FCAi � 0.970FCAj

FCD D 0.928FCDi C 0.371FCDj

For equilibrium, FCA C FCB C FCD D 0

In component form, this gives

∑Fx D 0.243FCA C 0.928FCD � 6.25 (kN) D 0

∑Fy D �0.970FCA C 0.371FCD C 5.00 (kN) D 0

Solving, we get

FCA D 7.02 kN, FCD D 4.89 kN

y

C

x

D

BA

FBC

FCD

FCA


115

Problem 3.48 The 50-lb cylinder rests on two smoothsurfaces.

(a) Draw the free-body diagram of the cylinder.(b) If ˛ D 30°, what are the magnitudes of the forces

exerted on the cylinder by the left and rightsurfaces? α 45°

Solution: Isolate the cylinder. (a) The free body diagram of theisolated cylinder is shown. (b) The forces acting are the weight and thenormal forces exerted by the surfaces. The angle between the normalforce on the right and the x axis is �90 C ˇ�. The normal force is

NR D jNRj�i cos�90 C ˇ� C j sin�90 C ˇ��

NR D jNRj��i sin ˇ C j cos ˇ�.

The angle between the positive x axis and the left hand force is normal�90 � ˛�; the normal force is NL D jNL j�i sin ˛ C j cos ˛�. The weightis W D 0i � jWjj. The equilibrium conditions are

∑F D W C NR C NL D 0.


∑Fx D ��jNRj sin ˇ C jNL j sin ˛�i D 0,

∑Fy D �jNRj cos ˇ C jNL j cos ˛ � jWj�j D 0.

y

xW

βα

NLNR

Solve: jNRj D(

sin ˛

sin ˇ

)jNL j,

and jNLj D( jWj sin ˇ

sin�˛ C ˇ�

).

For jWj D 50 lb, and ˛ D 30°, ˇ D 45°, the normal forces are

jNL j D 36.6 lb, jNRj D 25.9 lb

Problem 3.49 For the 50-lb cylinder in Problem 3.48,obtain an equation for the force exerted on the cylinderby the left surface in terms of the angle ˛ in two ways:(a) using a coordinate system with the y axis vertical,(b) using a coordinate system with the y axis parallel tothe right surface.

Solution: The solution for Part (a) is given in Problem 3.48 (seefree body diagram).

jNRj D(

sin ˛

sin ˇ

)jNL j jNL j D

( jWj sin ˇ

sin�˛ C ˇ�

).

Part (b): The isolated cylinder with the coordinate system is shown.The angle between the right hand normal force and the positive x axisis 180°. The normal force: NR D �jNRji C 0j. The angle between theleft hand normal force and the positive x is 180 � �˛ C ˇ�. The normalforce is NL D jNL j��i cos�˛ C ˇ� C j sin�˛ C ˇ��.

The angle between the weight vector and the positive x axis is �ˇ.The weight vector is W D jWj�i cos ˇ � j sin ˇ�.The equilibrium conditions are

∑F D W C NR C NL D 0.

y

x

α β

NL

NR

W


∑Fx D ��jNRj � jNL j cos�˛ C ˇ� C jWj cos ˇ�i D 0,

∑Fy D �jNL j sin�˛ C ˇ� � jWj sin ˇ�j D 0.

Solve: jNL j D( jWj sin ˇ

sin�˛ C ˇ�

)

116


Problem 3.50 The two springs are identical, withunstretched length 0.4 m. When the 50-kg mass issuspended at B, the length of each spring increases to0.6 m. What is the spring constant k?

C

B

A

k

k

0.6 m

Solution:

F D k�0.6 m � 0.4 m�

∑Fy : 2F sin 60° � 490.5 N D 0

k D 1416 N/m

60° 60°

FF

490.5 N

Problem 3.51 The cable AB is 0.5 m in length. Theunstretched length of the spring is 0.4 m. When the50-kg mass is suspended at B, the length of the springincreases to 0.45 m. What is the spring constant k?

C

B

A

k

0.7 m

Solution: The Geometry

Law of Cosines and Law of Sines

0.72 D 0.52 C 0.452 � 2�0.5��0.45� cos ˇ

sin �

0.45 mD sin �

0.5 mD sin ˇ

0.7 m

ˇ D 94.8°, � D 39.8° � D 45.4°

Now do the statics

F D k�0.45 m � 0.4 m�

∑Fx : �TAB cos � C F cos � D 0

∑Fy : TAB sin � C F sin � � 490.5 N D 0

Solving: k D 7560 N/m

θ φ

β

0.45 m0.5 m

0.7 m

θ φ

490.5 N

FTAB


117

Problem 3.52 The small sphere of mass m is attachedto a string of length L and rests on the smooth surfaceof a fixed sphere of radius R. The center of the sphereis directly below the point where the string is attached.Obtain an equation for the tension in the string in termsof m, L, h, and R.

R

h L

m

Solution: From the geometry we have

cos � D R C h � y

L, sin � D x

L

cos � D y

R, sin � D x

R

Thus the equilibrium equations can be written

∑Fx : � x

LT C x

RN D 0

∑Fy :

R C h � y

LT C y

RN � mg D 0

Solving, we find T D mgL

R C h

Problem 3.53 The inclined surface is smooth. Deter-mine the force T that must be exerted on the cable tohold the 100-kg crate in equilibrium and compare youranswer to the answer of Problem 3.11.

T

60�

Solution:∑

F- : 3 T � 981 N sin 60° D 0

T D 283 N

60°N

981 N

3T

118


Problem 3.54 In Example 3.3, suppose that the massof the suspended object is mA and the masses of thepulleys are mB D 0.3mA, mC D 0.2mA, and mD D 0.2mA.Show that the force T necessary for the system to be inequilibrium is 0.275mAg.

T

A

BBB

DD

CC

Solution: From the free-body diagram of pulley C

TD � 2T � mCg D 0 ) TD D 2T C mCg

Then from the free-body diagram of pulley B

T C T C 2T C mCg � mBg � mAg D 0

Thus

T D 1

4�mA C mB � mC�g

T D 1

4�mA C 0.3mA � 0.2mA�g D 0.275mAg

T D 0.275mAg

TT T

TD

T T

(a)

TD � 2T � mg

(b)

mAg

A

D

CC

C

B

B

mg

mg


119

Problem 3.55 The mass of each pulley of the systemis m and the mass of the suspended object A is mA.Determine the force T necessary for the system to be inequilibrium.

TA

Solution: Draw free body diagrams of each pulley and the objectA. Each pulley and the object A must be in equilibrium. The weightsof the pulleys and object A are W D mg and WA D mAg. The equilib-rium equations for the weight A, the lower pulley, second pulley, thirdpulley, and the top pulley are, respectively, B � WA D 0, 2C � B �W D 0, 2D � C � W D 0, 2T � D � W D 0, and FS � 2T � W D 0.Begin with the first equation and solve for B, substitute for B in thesecond equation and solve for C, substitute for C in the third equationand solve for D, and substitute for D in the fourth equation and solvefor T, to get T in terms of W and WA. The result is

B D WA, C D WA

2C W

2,

D D WA

4C 3W

4, and T D WA

8C 7W

8,

or in terms of the masses,

T D g

8�mA C 7m�.

Fs

WA

TT T

TWW

WC

CC

B

B

W

DDD

120


Problem 3.56 The suspended mass m1 D 50 kg. Neg-lecting the masses of the pulleys, determine the valueof the mass m2 necessary for the system to be inequilibrium.

A

B

C

m1

m2

Solution:∑

FC : T1 C 2m2g � m1g D 0

∑FB : T1 � 2m2g D 0

m2 D m1

4D 12.5 kg

B

T1

T = m2 gT

m1 g

TT1

T

C


121

Problem 3.57 The boy is lifting himself using theblock and tackle shown. If the weight of the block andtackle is negligible, and the combined weight of the boyand the beam he is sitting on is 120 lb, what forcedoes he have to exert on the rope to raise himself ata constant rate? (Neglect the deviation of the ropes fromthe vertical.)

Solution: A free-body diagram can be obtained by cutting the fourropes between the two pulleys of the block and tackle and the ropethe boy is holding. The tension has the same value T in all five ofthese ropes. So the upward force on the free-body diagram is 5T andthe downward force is the 120-lb weight. Therefore the force the boymust exert is

T D �120 lb�/5 D 24 lb

T D 24 lb

122


Problem 3.58 Pulley systems containing one, two, andthree pulleys are shown. Neglecting the weights of thepulleys, determine the force T required to support theweight W in each case.

T T T

W

W

W

(a) One pulley

(b) Two pulleys

(c) Three pulleys

Solution:

(a)∑

Fy : 2T � W D 0 ) T D W

2

(b)∑

Fupper : 2T � T1 D 0

∑Flower : 2T1 � W D 0

T D W

4

(c)∑

Fupper : 2T � T1 D 0

∑Fmiddle : 2T1 � T2 D 0

∑Flower : 2T2 � W D 0

T D W

8

(a) For one pulleys

W

T T

(b) For two pulleys

W

TT

T1 T1

(c) For three pulleys

TT

T1T1

T2T2

T2

W


123

Problem 3.59 Problem 3.58 shows pulley systemscontaining one, two, and three pulleys. The number ofpulleys in the type of system shown could obviously beextended to an arbitrary number N.

(a) Neglecting the weights of the pulleys, determinethe force T required to support the weight W as afunction of the number of pulleys N in the system.

(b) Using the result of part (a), determine the force Trequired to support the weight W for a system with10 pulleys.

Solution: By extrapolation of the previous problem

(a) T D W

2N

(b) T D W

1024

Problem 3.60 A 14,000-kg airplane is in steady flightin the vertical plane. The flight path angle is � D 10°,the angle of attack is ˛ D 4°, and the thrust force exertedby the engine is T D 60 kN. What are the magnitudesof the lift and drag forces acting on the airplane? (SeeExample 3.4).

Solution: Let us draw a more detailed free body diagram to seethe angles involved more clearly. Then we will write the equations ofequilibrium and solve them.

W D mg D �14,000��9.81� N

The equilibrium equations are

∑Fx D T cos ˛ � D � W sin � D 0

∑Fy D T sin ˛ C L � W cos � D 0

T D 60 kN D 60000 N

Solving, we get

D D 36.0 kN, L D 131.1 kN

y

TL

D

W

Horizon

γα

Path

x

yα = 4°γ = 10°

L

T

D

x

W

γ

α

γ

124


Problem 3.61 An airplane is in steady flight, the angleof attack ˛ D 0, the thrust-to-drag ratio T/D D 2, andthe lift-to-drag ratio L/D D 4. What is the flight pathangle �? (See Example 3.4).

Solution: Use the same strategy as in Problem 3.52. The anglebetween the thrust vector and the positive x axis is ˛,

T D jTj�i cos ˛ C j sin ˛�

The lift vector: L D 0i C jLjj

The drag: D D �jDji C 0j. The angle between the weight vector andthe positive x axis is �270 � ��;

W D jWj��i sin � � j cos ��.


∑F D T C L C D C W D 0.


∑Fx D �jTj cos ˛ � jDj � jWj sin ��i D 0,

and∑

Fy D �jTj sin ˛ C jLj � jWj cos ��j D 0

Solve the equations for the terms in � :

jWj sin � D jTj cos ˛ � jDj,

and jWj cos � D jTj sin ˛ C jLj.

Take the ratio of the two equations

tan � D( jTj cos ˛ � jDj

jTj sin ˛ C jLj)

.

Divide top and bottom on the right by jDj.

For ˛ D 0,jTjjDj D 2,

jLjjDj D 4, tan � D

(2 � 1

4

)D 1

4or � D 14°

y

xL

T

D

W

αγ

Path

Horizontal


125

Problem 3.62 An airplane glides in steady flight (T D0), and its lift-to-drag ratio is L/D D 4.

(a) What is the flight path angle �?(b) If the airplane glides from an altitude of 1000 m

to zero altitude, what horizontal distance does ittravel?

(See Example 3.4.)

Solution: See Example 3.4. The angle between the thrust vectorand the positive x axis is ˛:

T D jTj�i cos ˛ C j sin ˛�.

The lift vector: L D 0i C jLjj.

The drag: D D �jDji C 0j. The angle between the weight vector andthe positive x axis is �270 � ��:

W D jWj��i sin � � j cos ��.


∑F D T C L C D C W D 0.

Substitute and collect like terms:

∑Fx D �jTj cos ˛ � jDj � jWj sin ��i D 0

∑Fy D �jTj sin ˛ C jLj � jWj cos ��j D 0

Solve the equations for the terms in � ,

jWj sin � D jTj cos ˛ � jDj,

and jWj cos � D jTj sin ˛ C jLj

Part (a): Take the ratio of the two equilibrium equations:

tan � D( jTj cos ˛ � jDj

jTj sin ˛ C jLj)

.

Divide top and bottom on the right by jDj.

For ˛ D 0, jTj D 0,jLjjDj D 4, tan � D

(�1

4

)� D �14°

Part (b): The flight path angle is a negative angle measured from thehorizontal, hence from the equality of opposite interior angles the angle� is also the positive elevation angle of the airplane measured at thepoint of landing.

tan � D 1

h, h D 1

tan �D 1(

1

4

) D 4 km

y

xL

T

D

W

αγ

Path

Horizontal

1 km

h

γ

γ

126


Problem 3.63 In Active Example 3.5, suppose that theattachment point B is moved to the point (5,0,0) m. Whatare the tensions in cables AB, AC, and AD?

(�3, 0, 3) m

(�2, 0, �2) m

z

D

(0, �4, 0) m

(4, 0, 2) mB

A

100 kg

yC

x

Solution: The position vector from point A to point B can be usedto write the force TAB.

rAB D �5i C 4j� m

TAB D TABrAB

jrABj D TAB�0.781i C 0.625j�

Using the other forces from Active Example 3.5, we have

∑Fx : 0.781TAB � 0.408TAC � 0.514TAD D 0

∑Fy : 0.625TAB C 0.816TAC C 0.686TAD � 981 N D 0

∑Fz : �0.408TAC C 0.514TAD D 0

Solving yields TAB D 509 N, TAC D 487 N, TAD D 386 N

Problem 3.64 The force F D 800i C 200j (lb) acts atpoint A where the cables AB, AC, and AD are joined.What are the tensions in the three cables?

y

z

(12, 4, 2) ft

(0, 4, 6) ft

(6, 0, 0) ftB

C

A

F(0, 6, 0) ftD

x

Solution: We first write the position vectors

rAB D ��6i � 4j � 2k� ft

rAC D ��12i C 6k� ft

rAD D ��12i C 2j � 2k� ft

Now we can use these vectors to define the force vectors

TAB D TABrAB

jrABj D TAB��0.802i � 0.535j � 0.267k�

TAC D TACrAC

jrACj D TAC��0.949i C 0.316k�

TAD D TADrAD

jrADj D TAD��0.973i C 0.162j � 0.162k�

The equilibrium equations are then

∑Fx : �0.802TAB � 0.949TAC � 0.973TAD C 800 lb D 0

∑Fy : �0.535TAB C 0.162TAD C 200 lb D 0

∑Fz : �0.267TAB C 0.316TAC � 0.162TAD D 0

Solving, we find TAB D 405 lb, TAC D 395 lb, TAD D 103 lb


127

Problem 3.65 Suppose that you want to apply a1000-lb force F at point A in a direction such that theresulting tensions in cables AB, AC, and AD are equal.Determine the components of F.

y

z

(12, 4, 2) ft

(0, 4, 6) ft

(6, 0, 0) ftB

C

A

F(0, 6, 0) ftD

x

Solution: We first write the position vectors

rAB D ��6i � 4j � 2k� ft

rAC D ��12i C 6k� ft

rAD D ��12i C 2j � 2k� ft

Now we can use these vectors to define the force vectors

TAB D TrAB

jrABj D T��0.802i � 0.535j � 0.267k�

TAC D TrAC

jrACj D T��0.949i C 0.316k�

TAD D TrAD

jrADj D T��0.973i C 0.162j � 0.162k�

The force F can be written F D �Fxi C Fyj C Fzk�The equilibrium equations are then

∑Fx : �0.802T � 0.949T � 0.973T C Fx D 0 ) Fx D 2.72T

∑Fy : �0.535T C 0.162T C Fy D 0 ) Fy D 0.732T

∑Fz : �0.267T C 0.316T � 0.162T C Fz D 0 ) Fz D 0.113T

We also have the constraint equation√

Fx2 C Fy

2 C Fz2 D 1000 lb

) T D 363 lb

Solving, we find Fx D 990 lb, Fy D 135 lb, Fz D 41.2 lb

128


Problem 3.66 The 10-lb metal disk A is supported bythe smooth inclined surface and the strings AB and AC.The disk is located at coordinates (5,1,4) ft. What arethe tensions in the strings?

(0, 6, 0) ft

(8, 4, 0) ftC

y

B

z

2 ft

8 ft

10 ft

Ax

Solution: The position vectors are

rAB D ��5i C 5j � 4k� ft

rAC D �3i C 3j � 4k� ft

The angle ˛ between the inclined surface the horizontal is

˛ D tan�1�2/8� D 14.0°

We identify the following force:

TAB D TABrAB

jrABj D TAB��0.615i C 0.615j � 0.492k�

TAC D TACrAC

jrACj D TAC�0.514i C 0.514j � 0.686k�

N D N�cos ˛j C sin ˛k� D N�0.970j C 0.243k�

W D ��10 lb�j

The equilibrium equations are then

∑Fx : �0.615TAB C 0.514TAC D 0

∑Fy : 0.615TAB C 0.514TAC C 0.970N � 10 lb D 0

∑Fz : �0.492TAB � 0.686TAC C 0.243N D 0

Solving, we find N D 8.35 lb TAB D 1.54 lb, TAC D 1.85 lb


129

Problem 3.67 The bulldozer exerts a force FD2i (kip)at A. What are the tensions in cables AB, AC, and AD?

y

C

A

Dz4 ft

3 ft

2 ft

x

8 ft

8 ft

6 ft

B

Solution: Isolate the cable juncture. Express the tensions in termsof unit vectors. Solve the equilibrium equations. The coordinates ofpoints A, B, C, D are:

A�8, 0, 0�, B�0, 3, 8�, C�0, 2, �6�, D�0, �4, 0�.

The radius vectors for these points are

rA D 8i C 0j C 0k, rB D 0i C 3j C 8k,

rC D 0i C 2j � 6k, rD D 0i C 4j C 0k.

By definition, the unit vector parallel to the tension in cable AB is

eAB D rB � rA

jrB � rAj .

Carrying out the operations for each of the cables, the results are:

eAB D �0.6835i C 0.2563j C 0.6835k,

eAC D �0.7845i C 0.1961j � 0.5883k,

eAD D �0.8944i � 0.4472j C 0k.

The tensions in the cables are expressed in terms of the unit vectors,

TAB D jTABjeAB, TAC D jTACjeAC, TAD D jTADjeAD.

The external force acting on the juncture is F D 2000i C 0j C 0k. Theequilibrium conditions are

∑F D 0 D TAB C TAC C TAD C F D 0.

Substitute the vectors into the equilibrium conditions:

∑Fx D ��0.6835jTABj� 0.7845jTACj� 0.8944jTADjC2000�iD0

∑Fy D �0.2563jTABj C 0.1961jTACj � 0.4472jTADj�j D 0

∑Fz D �0.6835jTABj � 0.5883jTACj C 0jTADj�k D 0

The commercial program TK Solver Plus was used to solve theseequations. The results are

jTABj D 780.31 lb , jTACj D 906.49 lb , jTADj D 844.74 lb .

130


Problem 3.68 Prior to its launch, a balloon carrying aset of experiments to high altitude is held in place bygroups of student volunteers holding the tethers at B, C,and D. The mass of the balloon, experiments package,and the gas it contains is 90 kg, and the buoyancy forceon the balloon is 1000 N. The supervising professorconservatively estimates that each student can exert atleast a 40-N tension on the tether for the necessary lengthof time. Based on this estimate, what minimum numbersof students are needed at B, C, and D?

y

z Bx

C (10,0, –12) m

(16, 0, 16) m

D(–16, 0, 4) m

(0, 8, 0) mA

Solution:∑

Fy D 1000 � �90��9.81� � T D 0

T D 117.1 N

A�0, 8, 0�

B�16, 0, 16�

C�10, 0, �12�

D��16, 0, 4�

We need to write unit vectors eAB, eAC, and eAD.

eAB D 0.667i � 0.333j C 0.667k

eAC D 0.570i � 0.456j � 0.684k

eAD D �0.873i � 0.436j C 0.218k

We now write the forces in terms of magnitudes and unit vectors

FAB D 0.667FABi � 0.333FABj C 0.667FABk

FAC D 0.570FACi � 0.456FACj � 0.684FACk

FAD D �0.873FADi � 0.436FACj C 0.218FACk

T D 117.1j (N)

The equations of equilibrium are

∑Fx D 0.667FAB C 0.570FAC � 0.873FAD D 0

∑Fy D �0.333FAB � 0.456FAC � 0.436FAC C 117.1 D 0

∑Fz D 0.667FAB � 0.684FAC C 0.218FAC D 0

Solving, we get

FAB D 64.8 N ¾ 2 students

FAC D 99.8 N ¾ 3 students

FAD D 114.6 N ¾ 3 students

yT

(0, 8, 0)

(−16, 0, 4)

(10, 0, −12) m

A FAC

FAD

D

z

x

C

(16, 0, 16) mB

1000 N

(90) g

T


131

Problem 3.69 The 20-kg mass is suspended by cablesattached to three vertical 2-m posts. Point A is at(0, 1.2, 0) m. Determine the tensions in cables AB, AC,and AD.

B

A

C

D

y

zx

2 m0.3 m

1 m

1 m

Solution: Points A, B, C, and D are located at

A�0, 1.2, 0�, B��0.3, 2, 1�,

C�0, 2, �1�, D�2, 2, 0�

Write the unit vectors eAB, eAC, eAD

eAB D �0.228i C 0.608j C 0.760k

eAC D 0i C 0.625j � 0.781k

eAD D 0.928i C 0.371j C 0k

The forces are

FAB D �0.228FABi C 0.608FABj C 0.760FABk

FAC D 0FACi C 0.625FACj � 0.781FACk

FAD D 0.928FADi C 0.371FADj C 0k

W D ��20��9.81�j

The equations of equilibrium are

∑Fx D �0.228FAB C 0 C 0.928FAD D 0

∑Fy D 0.608FAB C 0.625FAC C 0.371FAD � 20�9.81� D 0

∑Fz D 0.760FAB � 0.781FAC C 0 D 0

We have 3 eqns in 3 unknowns solving, we get

FAB D 150.0 N

FAC D 146.1 N

FAD D 36.9 N

yC

A

FAD

FAC

FAB

D

W

B

zx

(20) (9.81) N

132


Problem 3.70 The weight of the horizontal wall sec-tion is W D 20,000 lb. Determine the tensions in thecables AB, AC, and AD.

CB6 ft

10 ft

4 ft 8 ft

7 ft

14 ft

W

D

A

Solution: Set the coordinate origin at A with axes as shown. Theupward force, T, at point A will be equal to the weight, W, since thecable at A supports the entire wall. The upward force at A is T D Wk. From the figure, the coordinates of the points in feet are

A�4, 6, 10�, B�0, 0, 0�, C�12, 0, 0�, and D�4, 14, 0�.

The three unit vectors are of the form

eAI D �xI � xA�i C �yI � yA�j C �zI � zA�k√�xI � xA�2 C �yI � yA�2 C �zI � zA�2

,

where I takes on the values B, C, and D. The denominators of the unitvectors are the distances AB, AC, and AD, respectively. Substitutionof the coordinates of the points yields the following unit vectors:

eAB D �0.324i � 0.487j � 0.811k,

eAC D 0.566i � 0.424j � 0.707k,

and eAD D 0i C 0.625j � 0.781k.

The forces are

TAB D TABeAB, TAC D TACeAC, and TAD D TADeAD.

The equilibrium equation for the knot at point A is

T C TAB C TAC C TAD D 0.

From the vector equilibrium equation, write the scalar equilibriumequations in the x, y, and z directions. We get three linear equationsin three unknowns. Solving these equations simultaneously, we get

TAB D 9393 lb, TAC D 5387 lb, and TAD D 10,977 lb

CB6 ft

10 ft

4 ft 8 ft

7 ft

14 ft

W

D

A

T

W

X

D

B

A

C

z

TB

TD

TC

y7 ft

14 ft

4 ft 8 ft

6 ft

10 ft


133

Problem 3.71 The car in Fig. a and the palletsupporting it weigh 3000 lb. They are supported byfour cables AB, AC, AD, and AE. The locations of theattachment points on the pallet are shown in Fig. b.The tensions in cables AB and AE are equal. Determinethe tensions in the cables.

C

A

y

B

E

(a)

x

8 ft 6 ft

6 ft 5 ft

5 ft

5 ft4 ft

D C

B

z

z

x

E

(b)

(0, 10, 0) ft

Solution: Isolate the knot at A. Let TAB, TAC, TAD and TAE bethe forces exerted by the tensions in the cables. The force exerted bythe vertical cable is (3000 lb)j. We first find the position vectors andthen express all of the forces as vectors.

rAB D �5i � 10j C 5k� ft

rAC D �6i � 10j � 5k� ft

rAD D ��8i � 10j � 4k� ft

rAE D ��6i � 10j C 5k� ft

TAB D TABrAB

jrABj D TAB�0.408i � 0.816j C 0.408k�

TAC D TACrAC

jrACj D TAC�0.473i � 0.788j � 0.394k�

TAD D TADrAD

jrADj D TAD��0.596i � 0.745j � 0.298k�

TAE D TAErAE

jrAEj D TAE��0.473i � 0.788j C 0.394k�

The equilibrium equations are

∑Fx : 0.408TAB C 0.473TAC � 0.596TAD � 0.473TAE D 0

∑Fy : �0.816TAB � 0.788TAC � 0.745TAD � 0.788TAE C 3000 lb D 0

∑Fz : 0.408TAB � 0.394TAC � 0.298TAD C 0.394TAE D 0

Solving, we find

TAB D 896 lb, TAC D 1186 lb, TAD D 843 lb, TAE D 896 lb

134


Problem 3.72 The 680-kg load suspended from thehelicopter is in equilibrium. The aerodynamic drag forceon the load is horizontal. The y axis is vertical, and cableOA lies in the x-y plane. Determine the magnitude of thedrag force and the tension in cable OA.

y A

10°

xO

BC D

Solution:∑

Fx D TOA sin 10° � D D 0,

∑Fy D TOA cos 10° � �680��9.81� D 0.

Solving, we obtain D D 1176 N, TOA D 6774 N.

y

xD

10°

TOA

(680) (9.81) N


135

Problem 3.73 In Problem 3.72, the coordinates of thethree cable attachment points B, C, and D are (�3.3,�4.5, 0) m, (1.1, �5.3, 1) m, and (1.6, �5.4, �1) m,respectively. What are the tensions in cables OB, OC,and OD?

Solution: The position vectors from O to pts B, C, and D are

rOB D �3.3i � 4.5j (m),

rOC D 1.1i � 5.3j C k (m),

rOD D 1.6i � 5.4j � k (m).

Dividing by the magnitudes, we obtain the unit vectors

eOB D �0.591i � 0.806j,

eOC D 0.200i � 0.963j C 0.182k,

eOD D 0.280i � 0.944j � 0.175k.

Using these unit vectors, we obtain the equilibrium equations

∑Fx D TOA sin 10° � 0.591TOB C 0.200TOC C 0.280TOD D 0,

∑Fy D TOA cos 10° � 0.806TOB � 0.963TOC � 0.944TOD D 0,

∑Fz D 0.182TOC � 0.175TOD D 0.

From the solution of Problem 3.72, TOA D 6774 N. Solving theseequations, we obtain

TOB D 3.60 kN, TOC D 1.94 kN, TOD D 2.02 kN.

y

x

TOA

TODTOC

TOB

10°

136


Problem 3.74 If the mass of the bar AB is negligiblecompared to the mass of the suspended object E, thebar exerts a force on the “ball” at B that points from Atoward B. The mass of the object E is 200 kg. The y-axispoints upward. Determine the tensions in the cables BCand CD.

Strategy: Draw a free-body diagram of the ball at B.(The weight of the ball is negligible.)

x

y

z

C

BD

A

(0, 5, 5) m

(0, 4, �3) m

(4, 3, 1) m

E

Solution:

FAB D FAB

(�4i � 3j � kp26

), TBC D TBC

(�4i C j � 4kp33

),

The forces

TBD D TBD

(�4i C 2j C 4k6

), W D ��200 kg��9.81 m/s2�j

The equilibrium equations

∑Fx : � 4p

26FAB � 4p

33TBC � 4

6TBD D 0

∑Fy : � 3p

26FAB C 1p

33TBC C 2

6TBD � 1962 N D 0

∑Fz : � 1p

26FAB � 4p

33TBC C 4

6TBD D 0

)TBC D 1610 N

TBD D 1009 N


137

Problem 3.75* The 3400-lb car is at rest on the planesurface. The unit vector en D 0.456i C 0.570j C 0.684kis perpendicular to the surface. Determine the magni-tudes of the total normal force N and the total frictionforce f exerted on the surface by the car’s wheels.

y

z

x

en

Solution: The forces on the car are its weight, the normal force,and the friction force.

The normal force is in the direction of the unit vector, so it can bewritten

N D Nen D N�0.456i C 0.570j C 0.684k�

The equilibrium equation is

Nen C f � �3400 lb�j D 0

The friction force f is perpendicular to N, so we can eliminate thefriction force from the equilibrium equation by taking the dot productof the equation with en.

�Nen C f � �3400 lb�j� Ð en D N � �3400 lb��j Ð en� D 0

N D �3400 lb��0.57� D 1940 lb

Now we can solve for the friction force f.

f D �3400 lb�j � Nen D �3400 lb�j � �1940 lb��0.456i C 0.570j C 0.684k�

f D ��884i C 2300j � 1330k� lb

jfj D√

��884 lb�2 C �2300 lb�2 C ��1330 lb�2 D 2790 lb

jNj D 1940 lb, jfj D 2790 lb

138


Problem 3.76 The system shown anchors a stanchionof a cable-suspended roof. If the tension in cable AB is900 kN, what are the tensions in cables EF and EG?

y

z

x

F

(0, 1.4, 1.2) m

(0, 1.4, �1.2) m

(1, 1.2, 0) m(2, 1, 0) m

(2.2, 0, 1) m

(2.2, 0, �1) m

(3.4, 1, 0) m

G

E

B A

C

D

Solution: Using the coordinates for the points we find

rBA D [�3.4 � 2�i C �1 � 1�j C �0 � 0�k] m

rBA D �1.4 m�i eBA D rBA

jrBAj D i

Using the same procedure we find the other unit vectors that we need.

eBC D 0.140i � 0.700j C 0.700k

eBD D 0.140i � 0.700j C 0.700k

eBE D �0.981i C 0.196j

eEG D �0.635i C 0.127j � 0.762k

eEF D �0.635i C 0.127j C 0.762k

We can now write the equilibrium equations for the connections at Band E.

�900 kN�eBA C TBCeBC C TBDeBD C TBEeBE D 0, TBE��eBE� C TEFeEF C TEGeEG0

Breaking these equations into components, we have the following sixequations to solve for five unknows (one of the equations is redundant).

�900 kN� C TBC�0.140� C TBD�0.140� C TBE��0.981� D 0

TBC��0.700� C TBD��0.700� C TBE�0.196� D 0

TBC�0.700� C TBD��0.700� D 0

TBE�0.981� C TEG��0.635� C TEF��0.635� D 0

TBE��0.196� C TEG�0.127� C TEF�0.127� D 0

TEG��0.726� C TEF�0.726� D 0

Solving, we find TBC D TBD D 134 kN, TBE D 956 kN

TEF D TEG D 738 kN

Problem 3.77* The cables of the system will eachsafely support a tension of 1500 kN. Based on thiscriterion, what is the largest safe value of the tensionin cable AB?

y

z

x

F

(0, 1.4, 1.2) m

(0, 1.4, �1.2) m

(1, 1.2, 0) m(2, 1, 0) m

(2.2, 0, 1) m

(2.2, 0, �1) m

(3.4, 1, 0) m

G

E

B A

C

D

Solution: From Problem 3.76 we know that if the tension in ABis 900 kN, then the largest force in the system occurs in cable BE andthat tension is 956 kN. To solve this problem, we can just scale theresults from Problem 3.76

TAB D(

1500 kN

956 kN

)�900 kN� TAB D 1410 kN


139

Problem 3.78 The 200-kg slider at A is held in placeon the smooth vertical bar by the cable AB.

(a) Determine the tension in the cable.(b) Determine the force exerted on the slider by

the bar.

z

y

x

A

B

2 m

5 m

2 m

2 m

Solution: The coordinates of the points A, B are A�2, 2, 0�,B�0, 5, 2�. The vector positions

rA D 2i C 2j C 0k, rB D 0i C 5j C 2k

The equilibrium conditions are:

∑F D T C N C W D 0.

Eliminate the slider bar normal force as follows: The bar is parallel tothe y axis, hence the unit vector parallel to the bar is eB D 0i C 1j C0k. The dot product of the unit vector and the normal force vanishes:eB Ð N D 0. Take the dot product of eB with the equilibrium conditions:eB Ð N D 0.

∑eB Ð F D eB Ð T C eB Ð W D 0.

The weight is

eB Ð W D 1j Ð ��jjWj� D �jWj D ��200��9.81� D �1962 N.

The unit vector parallel to the cable is by definition,

eAB D rB � rA

jrB � rAj .

Substitute the vectors and carry out the operation:

eAB D �0.4851i C 0.7278j C 0.4851k.

(a) The tension in the cable is T D jTjeAB. Substitute into the modi-fied equilibrium condition

∑eBF D �0.7276jTj � 1962� D 0.

Solve: jTj D 2696.5 N from which the tension vector is

T D jTjeAB D �1308i C 1962j C 1308k.

(b) The equilibrium conditions are

∑F D 0 D T C N C W D �1308i C 1308k C N D 0.

Solve for the normal force: N D 1308i � 1308k. The magnitudeis jNj D 1850 N.

T

N

W

Note: For this specific configuration, the problem can be solved with-out eliminating the slider bar normal force, since it does not appear inthe y-component of the equilibrium equation (the slider bar is parallelto the y-axis). However, in the general case, the slider bar will not beparallel to an axis, and the unknown normal force will be projectedonto all components of the equilibrium equations (see Problem 3.79below). In this general situation, it will be necessary to eliminate theslider bar normal force by some procedure equivalent to that usedabove. End Note.

140


Problem 3.79 In Example 3.6, suppose that the cableAC is replaced by a longer one so that the distance frompoint B to the slider C increases from 6 ft to 8 ft. Deter-mine the tension in the cable.

6 ft

O

y

C

A

B

Dz

4 ft

7 ft

4 ft4 ft

x

Solution: The vector from B to C is now

rBC D �8 ft� eBD

rBC D �8 ft�

(4

9i � 7

9j C 4

9k)

rBC D �3.56i � 6.22j C 3.56k� ft

We can now find the unit vector form C to A.

rCA D rOA � �rOB C rBC� D [�7j C 4k�

� f�7j� C �3.56i � 6.22j C 3.56k�g] ft

rCA D ��3.56i C 6.22j C 0.444k� ft

eCA D rCA

jrCAj D ��0.495i C 0.867j C 0.0619k�

Using N to stand for the normal force between the bar and the slider,we can write the equilibrium equation:

TeCA C N � �100 lb�j D 0

We can use the dot product to eliminate N from the equation

[TeCA C N � �100 lb�j] Ð eBD D T�eCA Ð eBD� � �100 lb��j Ð eBD� D 0

T

([4

9

][�0.495] C

[� 7

9

][0.867] C

[4

9

][0.0619]

)� �100 lb��0.778� D 0

T��0.867� C �77.8 lb� D 0 ) T D 89.8 lb

T D 89.8 lb


141

Problem 3.80 The cable AB keeps the 8-kg collar A inplace on the smooth bar CD. The y axis points upward.What is the tension in the cable?

0.4 m

0.5 m

0.15 m

0.3 m0.2 m

0.25 m

0.2 mz

y

BC

D

O x

A

Solution: We develop the following position vectors and unitvectors

rCD D ��0.2i � 0.3j C 0.25k� m

eCD D rCD

jrCDj D ��0.456i � 0.684j C 0.570k�

rCA D �0.2 m�eCD D ��0.091i � 0.137j C 0.114k� m

rAB D rOB � �rOC C rCA�

rAB D [�0.5j C 0.15k� � �f0.4i C 0.3jg C f�0.091i � 0.137j C 0.114kg�] m

rAB D ��0.309i C 0.337j C 0.036k� m

eAB D rAB

jrABj D ��0.674i C 0.735j C 0.079k�

We can now write the equilibrium equation for the slider using N tostand for the normal force between the slider and the bar CD.

TeAB C N � �8 kg��9.81 m/s2�j D 0

To eliminate the normal force N we take a dot product with eCD.

[TeAB C N � �8 kg��9.81 m/s2�j] Ð eCD D 0

T�eAB Ð eCD� � �78.5 N��j Ð eCD� D 0

T�[�0.674][�0.456] C [0.735][�0.684] C [0.079][0.570]� � �78.5 N��0.684� D 0

T��0.150� C 53.6 N D 0

T D 357 N

Problem 3.81 Determine the magnitude of the normalforce exerted on the collar A by the smooth bar.

0.4 m

0.5 m

0.15 m

0.3 m0.2 m

0.25 m

0.2 mz

y

BC

D

O x

A

Solution: From Problem 3.81 we have

eAB D ��0.674i C 0.735j C 0.079k�

T D 357 N


TeAB C N � �78.5 N�j D 0

We can now solve for the normal force N.

N D �78.5 N�j � �357 N��0.674i C 0.735j C 0.079k�

N D �240i � 184j � 28.1k� N

The magnitude of N is

jNj D√

�240 N�2 C ��184 N�2 C ��28.1 N�2

jNj D 304 N

142


Problem 3.82* The 10-kg collar A and 20-kg collar Bare held in place on the smooth bars by the 3-m cablefrom A to B and the force F acting on A. The force Fis parallel to the bar. Determine F.

z

x

y

B

A

F

3 m

(4, 0, 0) m

(0, 0, 4) m

(0, 3, 0) m

(0, 5, 0) m

Solution: The geometry is the first part of the Problem. To easeour work, let us name the points C, D, E, and G as shown in thefigure. The unit vectors from C to D and from E to G are essential tothe location of points A and B. The diagram shown contains two freebodies plus the pertinent geometry. The unit vectors from C to D andfrom E to G are given by

eCD D erCDx i C eCDyj C eCDzk,

and eEG D erEGx i C eEGyj C eEGzk.

Using the coordinates of points C, D, E, and G from the picture, theunit vectors are

eCD D �0.625i C 0.781j C 0k,

and eEG D 0i C 0.6j C 0.8k.

The location of point A is given by

xA D xC C CAeCDx, yA D yC C CAeCDy,

and zA D zC C CAeCDz,

where CA D 3 m. From these equations, we find that the location ofpoint A is given by A (2.13, 2.34, 0) m. Once we know the locationof point A, we can proceed to find the location of point B. We havetwo ways to determine the location of B. First, B is 3 m from point Aalong the line AB (which we do not know). Also, B lies on the lineEG. The equations for the location of point B based on line AB are:

xB D xA C ABeABx, yB D yA C ABeABy,

and zB D zA C ABeABz.

The equations based on line EG are:

xB D xE C EBeEGx, yB D yE C EBeEGy,

and zB D zE C EBeEGz.

We have six new equations in the three coordinates of B and thedistance EB. Some of the information in the equations is redundant.However, we can solve for EB (and the coordinates of B). We get thatthe length EB is 2.56 m and that point B is located at (0, 1.53, 1.96) m.We next write equilibrium equations for bodies A and B. From the freebody diagram for A, we get

NAx C TABeABx C FeCDx D 0,

NAy C TABeABy C FeCDy � mAg D 0,

and NAz C TABeABz C FeCDz D 0.

From the free body diagram for B, we get

NBx � TABeABx D 0,

Nby � TABeABy � mBg D 0,

and NBz � TABeABz D 0.

y

F

B x

z

3 m

NA

TAB

TABA

NB

D (0, 5, 0)

C (4, 0, 0) m

E (0, 0, 4) m

m

G (0, 3, 0)m

mAgmBg

We now have two fewer equation than unknowns. Fortunately, thereare two conditions we have not yet invoked. The bars at A and Bare smooth. This means that the normal force on each bar can haveno component along that bar. This can be expressed by using the dotproduct of the normal force and the unit vector along the bar. The twoconditions are

NA Ð eCD D NAxeCDx C NAyeCDy C NAzeCDz D 0

for slider A and

NB Ð eEG D NBxeEGx C NByeEGy C NBzeEGz D 0.

Solving the eight equations in the eight unknowns, we obtain

F D 36.6 N .

Other values obtained in the solution are EB D 2.56 m,

NAx D 145 N, NAy D 116 N, NAz D �112 N,

NBx D �122 N, NBy D 150 N, and NBz D 112 N.


143

Problem 3.83 The 100-lb crate is held in place on thesmooth surface by the rope AB. Determine the tension inthe rope and the magnitude of the normal force exertedon the crate by the surface.

45�

A

B

30�

Solution: The free-body diagram is sketched. The equilibriumequations are

∑Fx- : T cos�45° � 30°� � �100 lb� sin 30° D 0

∑F% : T sin�45° � 30°� � �100 lb� cos 30° C N D 0

Solving, we find

T D 51.8 lb, N D 73.2 lb

Problem 3.84 The system shown is called Russell’straction. If the sum of the downward forces exerted atA and B by the patient’s leg is 32.2 lb, what is the weightW?

y

W

A

B

20�

25�

60�

x

Solution: The force in the cable is W everywhere. The free-bodydiagram of the leg is shown. The downward force is given, but thehorizontal force FH is unknown.The equilibrium equation in the vertical direction is

∑Fy : W sin 25° C W sin 60° � 32.2 lb D 0

Thus

W D 32.2 lb

sin 25° C sin 60°

W D 25.0 lb

144


Problem 3.85 The 400-lb engine block is suspendedby the cables AB and AC. If you don’t want either TABor TAC to exceed 400 lb, what is the smallest acceptablevalue of the angle ˛?

BC

A Ax

y

a a

TAB TAC

400 lb


∑Fx : �TAB cos ˛ C TAC cos ˛ D 0

∑Fy : TAB sin ˛ C TAC sin ˛ � �400 lb� D 0

Solving, we find

TAB D TAC D 400 lb

2 sin ˛

If we limit the tensions to 400 lb, we have

400 lb D 400 lb

2 sin ˛) sin ˛ D 1

2) ˛ D 30°

Problem 3.86 The cable AB is horizontal, and the boxon the right weighs 100 lb. The surface are smooth.(a) What is the tension in the cable?(b) What is the weight of the box on the left?

A B

20�

40�

Solution: We have the following equilibrium equations

∑FyB : NB cos 40° � 100 lb D 0

∑FxB : NB sin 40° � T D 0

∑FxA : T � NA sin 20° D 0

∑FyA : NA cos 20° � WA D 0

Solving these equations sequentially, we find

NB D 131 lb, T D 83.9 lb

NA D 245 lb, WA D 230.5 lb

Thus we have T D 83.9 lb, WA D 230.5 lb


145

Problem 3.87 Assume that the forces exerted on the170-lb climber by the slanted walls of the “chimney”are perpendicular to the walls. If he is in equilibriumand is exerting a 160-lb force on the rope, what are themagnitudes of the forces exerted on him by the left andright walls?

4�

10�

3�

Solution: The forces in the free-body diagram are in the directionsshown on the figure. The equilibrium equations are:

∑Fx : �T sin 10° C NL cos 4° � NR cos 3° D 0

∑Fy : T cos 10° � �170 lb� C NL sin 40° C NR sin 3° D 0

where T D 160 lb. Solving we find

NL D 114 lb, NR D 85.8 lb

Left Wall: 114 lbRight Wall: 85.8 lb

146


Problem 3.88 The mass of the suspended object A ismA and the masses of the pulleys are negligible. Deter-mine the force T necessary for the system to be inequilibrium.

T A

Solution: Break the system into four free body diagrams as shown.Carefully label the forces to ensure that the tension in any single cordis uniform. The equations of equilibrium for the four objects, startingwith the leftmost pulley and moving clockwise, are:

S � 3T D 0, R � 3S D 0, F � 3R D 0,

and 2T C 2S C 2R � mAg D 0.

We want to eliminate S, R, and F from our result and find T interms of mA and g. From the first two equations, we get S D 3T,and R D 3S D 9T. Substituting these into the last equilibrium equationresults in 2T C 2�3T� C 2�9T� D mAg.

Solving, we get T D mAg/26 .

R

R

R

R R

R

F

SS

SS

SS

TT

TT

T

A

mAg

Note: We did not have to solve for F to find the appropriate value of T.The final equation would give us the value of F in terms of mA and g.We would get F D 27mAg/26. If we then drew a free body diagramof the entire assembly, the equation of equilibrium would be F � T �mAg D 0. Substituting in the known values for T and F, we see thatthis equation is also satisfied. Checking the equilibrium solution byusing the “extra” free body diagram is often a good procedure.


147

Problem 3.89 The assembly A, including the pulley,weighs 60 lb. What force F is necessary for the systemto be in equilibrium?

F

A

Solution: From the free body diagram of the assembly A, we have3F � 60 D 0, or F D 20 lb

FF

FF

F F

60 lb.

F

148


Problem 3.90 The mass of block A is 42 kg, and themass of block B is 50 kg. The surfaces are smooth. Ifthe blocks are in equilibrium, what is the force F?

B

45�

A

20�

F

Solution: Isolate the top block. Solve the equilibrium equations.The weight is. The angle between the normal force N1 and the posi-tive x axis is. The normal force is. The force N2 is. The equilibriumconditions are

∑F D N1 C N2 C W D 0

from which∑

Fx D �0.7071jN1j � jN2j�i D 0

∑Fy D �0.7071jN1j � 490.5�j D 0.

Solve: N1 D 693.7 N, jN2j D 490.5 N

Isolate the bottom block. The weight is

W D 0i � jWjj D 0i � �42��9.81�j D 0i � 412.02j (N).

The angle between the normal force N1 and the positive x axis is�270° � 45°� D 225°.

The normal force:

N1 D jN1j�i cos 225° C j sin 225°� D jN1j��0.7071i � 0.7071j�.

The angle between the normal force N3 and the positive x-axis is�90° � 20°� D 70°.

The normal force is

N1 D jN3j�i cos 70° C j sin 70°� D jN3j�0.3420i C 0.9397j�.

The force is . . . F D jFji C 0j. The equilibrium conditions are

∑F D W C N1 C N3 C F D 0,

from which:∑

Fx D ��0.7071jN1j C 0.3420jN3j C jFj�i D 0

∑Fy D ��0.7071jN1j C 0.9397jN3j � 412�j D 0

For jN1j D 693.7 N from above: jFj D 162 N

y

x

B

W

N2

N1 α

x

y

W

F A

N3

N1

α

β


149

Problem 3.91 The climber A is being helped up an icyslope by two friends. His mass is 80 kg, and the directioncosines of the force exerted on him by the slope arecos �x D �0.286, cos �y D 0.429, cos �z D 0.857. The yaxis is vertical. If the climber is in equilibrium in theposition shown, what are the tensions in the ropes ABand AC and the magnitude of the force exerted on himby the slope?

A(3, 0, 4) m

B(2, 2, 0) m

C(5, 2, �1) m

y

xz

Solution: Get the unit vectors parallel to the ropes using the coor-dinates of the end points. Express the tensions in terms of these unitvectors, and solve the equilibrium conditions. The rope tensions, thenormal force, and the weight act on the climber. The coordinatesof points A, B, C are given by the problem, A�3, 0, 4�, B�2, 2, 0�,C�5, 2, �1�.

The vector locations of the points A, B, C are:

rA D 3i C 0j C 4k, rB D 2i C 2j C 0k, rC D 5i C 2j � 1k.

The unit vector parallel to the tension acting between the points A, Bin the direction of B is

eAB D rB � rA

jrB � rAj

The unit vectors are

eAB D �0.2182i C 0.4364j � 0.8729k,

eAC D 0.3482i C 0.3482j � 0.8704k,

and eN D �0.286i C 0.429j C 0.857k.

where the last was given by the problem statement. The forces areexpressed in terms of the unit vectors,

TAB D jTABjeAB, TAC D jTACjeAC, N D jNjeN.

The weight is

W D 0i � jWjj C 0k D 0i � �80��9.81�j C 0k � 0i � 784.8j C 0k.


∑F D 0 D TAB C TAC C N C W D 0.

A

C

WN

B


∑Fx D ��0.2182jTABj C 0.3482jTACj � 0.286jNj�i D 0

∑Fy D �0.4364jTABj C 0.3482jTACj C 0.429jNj � 784.8�j D 0

∑Fz D �0.8729jTABj C 0.8704jTACj � 0.857jNj�k D 0

We have three linear equations in three unknowns. The solution is:

jTABj D 100.7 N , jTACj D 889.0 N , jNj D 1005.5 N .

150


Problem 3.92 Consider the climber A being helped byhis friends in Problem 3.91. To try to make the tensionsin the ropes more equal, the friend at B moves to theposition (4, 2, 0) m. What are the new tensions in theropes AB and AC and the magnitude of the force exertedon the climber by the slope?

Solution: Get the unit vectors parallel to the ropes using the coor-dinates of the end points. Express the tensions in terms of theseunit vectors, and solve the equilibrium conditions. The coordinatesof points A, B, C are A�3, 0, 4�, B�4, 2, 0�, C�5, 2, �1�. The vectorlocations of the points A, B, C are:

rA D 3i C 0j C 4k, rB D 4i C 2j C 0k, rC D 5i C 2j � 1k.

The unit vectors are

eAB D C0.2182i C 0.4364j � 0.8729k,

eAC D C0.3482i C 0.3482j � 0.8704k,

eN D �0.286i C 0.429j C 0.857k.

where the last was given by the problem statement. The forces areexpressed in terms of the unit vectors,

TAB D jTABjeAB, TAC D jTACjeAC, N D jNjeN.

The weight is

W D 0i � jWjj C 0k D 0i � �80��9.81�j C 0k � 0i � 784.8j C 0k.


∑F D 0 D TAB C TAC C N C W D 0.


∑Fx D �C0.281jTABj C 0.3482jTACj � 0.286jNj�i D 0

∑Fy D �0.4364jTABj C 0.3482jTACj C 0.429jNj � 784.8�j D 0

∑Fz D �0.8729jTABj C 0.8704jTACj � 0.857jNj�k D 0

The HP-28S hand held calculator was used to solve these simultaneousequations. The solution is:

jTABj D 420.5 N , jTACj D 532.7 N , jNj D 969.3 N .


151

Problem 3.93 A climber helps his friend up an icyslope. His friend is hauling a box of supplies. If themass of the friend is 90 kg and the mass of the suppliesis 22 kg, what are the tensions in the ropes AB and CD?Assume that the slope is smooth. That is, only normalforces are exerted on the man and the box by the slope.

D

60�

75

40�

C

B20�

A

Solution: Isolate the box. The weight vector is

W2 D �22��9.81�j D �215.8j (N).

The angle between the normal force and the positive x axis is �90° �60°� D 30°.

The normal force is NB D jNBj�0.866i � 0.5j�.

The angle between the rope CD and the positive x axis is �180° �75°� D 105°; the tension is:

T2 D jT2j�i cos 105° C j sin 105°� D jT2j��0.2588i C 0.9659j�


∑Fx D �0.866jNBj C 0.2588jT2j�i D 0,

∑Fy D �0.5jNBj C 0.9659jT2j � 215.8�j D 0.

Solve: NB D 57.8 N, jT2j D 193.5 N.

Isolate the friend. The weight is

W D ��90��9.81�j D �882.9j (N).

The angle between the normal force and the positive x axis is �90° �40°� D 50°. The normal force is:

NF D jNFj�0.6428i C 0.7660j�.

The angle between the lower rope and the x axis is �75°; the tension is

T2 D jT2j�0.2588i C 0.9659j�.

The angle between the tension in the upper rope and the positive xaxis is �180° � 20°� D 160° , the tension is

T1 D jT1j�0.9397i C 0.3420j�.


∑F D W C T1 C T2 C NF D 0.

From which:

∑Fx D �0.6428jNFj C 0.2588jT2j � 0.9397jT1j�i D 0

∑Fy D ��0.7660jNFj � 0.9659jT2j C 0.3420jT1j � 882.9�j D 0

Solve, for jT2j D 193.5 N. The result:

jNFj D 1051.6 N , jT1j D 772.6 N .

y

x

N

W

α

β

T

y

xWN

T1

T2

20°

40°75°

152


Problem 3.94 The 2800-lb car is moving at constantspeed on a road with the slope shown. The aerodynamicforces on the car the drag D D 270 lb, which is parallelto the road, and the lift L D 120 lb, which is perpendic-ular to the road. Determine the magnitudes of the totalnormal and friction forces exerted on the car by the road.

DL

15�

Solution: The free-body diagram is shown. If we write the equi-librium equations parallel and perpendicular to the road, we have:

∑F- : N � �2800 lb� cos 15° C �120 lb� D 0

∑F% : f � �270 lb� � �2800 lb� sin 15° D 0

Solving, we find

N D 2580 lb, f D 995 lb


153

Problem 3.95 An engineer doing preliminary designstudies for a new radio telescope envisions a triangularreceiving platform suspended by cables from three equ-ally spaced 40-m towers. The receiving platform hasa mass of 20 Mg (megagrams) and is 10 m below thetops of the towers. What tension would the cables besubjected to?

65 m

20 m

TOP VIEW

Solution: Isolate the platform. Choose a coordinate system withthe origin at the center of the platform, with the z axis vertical, andthe x,y axes as shown. Express the tensions in terms of unit vectors,and solve the equilibrium conditions. The cable connections at theplatform are labeled a, b, c, and the cable connections at the towersare labeled A, B, C. The horizontal distance from the origin (center ofthe platform) to any tower is given by

L D 65

2 sin�60�D 37.5 m.

The coordinates of points A, B, C are

A�37.5, 0, 10�, B�37.5 cos�120°�, 37.5 sin�120°�.10�,

C�37.5 cos�240°�, 37.5 sin�240°�, 10�,

The vector locations are:

rA D 37.5i C 0j C 10k, rB D 18.764i C 32.5j C 10k,

rC D 18.764i C �32.5j C 10k.

The distance from the origin to any cable connection on the platform is

d D 20

2 sin�60°�D 11.547 m.

The coordinates of the cable connections are

a�11.547, 0, 0�, b�11.547 cos�120°�, 11547 sin�120°�, 0�,

c�11.547 cos�240°�, 11.547 sin�240°�, 0�.

The vector locations of these points are,

ra D 11.547i C 0j C 0k, rb D 5.774i C 10j C 0k,

rc D 5.774i C 10j C 0k.

The unit vector parallel to the tension acting between the points A, ain the direction of A is by definition

eaA D rA � ra

jrA � ra.

Perform this operation for each of the unit vectors to obtain

eaA D C0.9333i C 0j � 0.3592k

ebB D �0.4667i C 0.8082j � 0.3592k

ecC D �0.4667i C 0.8082j C 0.3592k

A

C

B

x y

z

a b

c


TaA D jTaAjeaA, TbB D jTbBjebB, TcC D jTcCjecC.

The weight is W D 0i � 0j � �20000��9.81�k D 0i C 0j � 196200k.


∑F D 0 D TaA C TbB C TcC C W D 0,

from which:

∑Fx D �0.9333jTaAj � 0.4666jTbBj � 0.4666jTcCj�i D 0

∑Fy D �0jTaAj C 0.8082jTbBj � 0.8082jTcCj�j D 0

∑Fz D �0.3592jTaAj � 0.3592jTbBj

C 0.3592jTcC � 196200j�k D 0

The commercial package TK Solver Plus was used to solve theseequations. The results:

jTaAj D 182.1 kN , jTbBj D 182.1 kN , jTcCj D 182.1 kN .

Check: For this geometry, where from symmetry all cable tensions maybe assumed to be the same, only the z-component of the equilibriumequations is required:

∑Fz D 3jTj sin � � 196200 D 0,

where � D tan�1

(10

37.5 � 11.547

)D 21.07°,

from which each tension is jTj D 182.1 kN. check.

154


Problem 3.96 To support the tent, the tension in therope AB must be 35 lb. What are the tensions in theropes AC, AD, and AE?

x

A B

C

D

(0, 5, 0) ft

(6, 4, 3) ft

(8, 4, 3) ft

E (3, 0, 3) ft

(0, 6, 6) ft

z

y

Solution: We develop the following position vectors

rAB D �2i� ft

rAC D ��6i C j � 3k� ft

rAD D ��6i C 2j C 3k� ft

rAE D ��3i � 4j� ft

If we divide by the respective magnitudes we can develop the unitvectors that are parallel to these position vectors.

eAB D 1.00i

eAC D �0.885i C 0.147j � 0.442k

eAD D �0.857i C 0.286j C 0.429k

eAE D �6.00i � 0.800j


TABeAB C TACeAC C TADeAD C TAEeAE D 0.

If we break this up into components, we have

∑Fx : TAB � 0.885TAC � 0.857TAD � 0.600TAE D 0

∑Fy : 0.147TAC C 0.286TAD � 0.800TAE D 0

∑Fz : �0.442TAC C 0.429TAD D 0

If we set TAB D 35 lb, we cans solve for the other tensions. The resultis

TAC D 16.7 lb, TAD D 17.2 lb, TAE D 9.21 lb


155

Problem 3.97 Cable AB is attached to the top of thevertical 3-m post, and its tension is 50 kN. What are thetensions in cables AO, AC, and AD?

x

3 m

y

AB

D

O

C

(6, 2, 0) m

z

12 m

4 m

5 m

5 m

8 m

Solution: Get the unit vectors parallel to the cables using thecoordinates of the end points. Express the tensions in terms of theseunit vectors, and solve the equilibrium conditions. The coordinates ofpoints A, B, C, D, O are found from the problem sketch: The coordi-nates of the points are A�6, 2, 0�, B�12, 3, 0�, C�0, 8, 5�, D�0, 4, �5�,O�0, 0, 0�.

The vector locations of these points are:

rA D 6i C 2j C 0k, rB D 12i C 3j C 0k, rC D 0i C 8j C 5k,

rD D 0i C 4j � 5k, rO D 0i C 0j C 0k.

The unit vector parallel to the tension acting between the points A, Bin the direction of B is by definition

eAB D rB � rA

jrB � rAj .

Perform this for each of the unit vectors

eAB D C0.9864i C 0.1644j C 0k

eAC D �0.6092i C 0.6092j C 0.5077k

eAD D �0.7442i C 0.2481j � 0.6202k

eAO D �0.9487i � 0.3162j C 0k


TAB D jTABjeAB D 50eAB, TAC D jTACjeAC,

TAD D jTADjeAD, TAO D jTAOjeAO.


∑F D 0 D TAB C TAC C TAD C TAO D 0.


∑Fx D �0.9864�50� � 0.6092jTACj � 0.7422jTADj

� 0.9487jTAOj�i D 0

∑Fy D �0.1644�50� C 0.6092jTACj C 0.2481jTADj

� 0.3162jTAOj�j D 0

∑Fz D �C0.5077jTACj � 0.6202jTADj�k D 0.

This set of simultaneous equations in the unknown forces may besolved using any of several standard algorithms. The results are:

jTAOj D 43.3 kN, jTACj D 6.8 kN, jTADj D 5.5 kN.

y

A

D

O

C

(6, 2, 0) m

4 m

5 m

5 m

8 m

156


Problem 3.98* The 1350-kg car is at rest on aplane surface with its brakes locked. The unit vectoren D 0.231i C 0.923j C 0.308k is perpendicular to thesurface. The y axis points upward. The direction cosinesof the cable from A to B are cos �x D �0.816, cos �y D0.408, cos �z D �0.408, and the tension in the cable is1.2 kN. Determine the magnitudes of the normal andfriction forces the car’s wheels exert on the surface.

en

ep

y

z

x

B

Solution: Assume that all forces act at the center of mass of thecar. The vector equation of equilibrium for the car is

FS C TAB C W D 0.

Writing these forces in terms of components, we have

W D �mgj D ��1350��9.81� D �13240j N,

FS D FSx i C FSy j C FSzk,

and TAB D TABeAB,

where

eAB D cos �x i C cos �y j C cos �zk D �0.816i C 0.408j � 0.408k.

Substituting these values into the equations of equilibrium and solvingfor the unknown components of FS, we get three scalar equations ofequilibrium. These are:

FSx � TABx D 0, FSy � TABy � W D 0,

and FSz � TABz D 0.

Substituting in the numbers and solving, we get

FSx D 979.2 N, FSy D 12, 754 N,

and FSz D 489.6 N.

The next step is to find the component of FS normal to the surface.This component is given by

FN D FN Ð en D FSxeny C FSxeny C FSzenz.

Substitution yields

FN D 12149 N .

From its components, the magnitude of FS is FS D 12800 N. Usingthe Pythagorean theorem, the friction force is

f D√

F2S � F2

N D 4033 N.

z

x

y

W

en

FNFS

F

TAB

"car"


157

Problem 3.99* The brakes of the car in Problem 3.98are released, and the car is held in place on theplane surface by the cable AB. The car’s front wheelsare aligned so that the tires exert no friction forcesparallel to the car’s longitudinal axis. The unit vectorep D �0.941i C 0.131j C 0.314k is parallel to the planesurface and aligned with the car’s longitudinal axis.What is the tension in the cable?

Solution: Only the cable and the car’s weight exert forces in thedirection parallel to ep. Therefore

ep Ð �T � mgj� D 0: ��0.941i C 0.131j C 0.314k�

Ð [T��0.816i C 0.408j � 0.408k� � mgj] D 0,

�0.941��0.816�T

C �0.131��0.408T � mg� C �0.314��0.408T� D 0.

Solving, we obtain T D 2.50 kN.

158


Engineering

Statics bedford 5 chap 03